902 * constexpr LowStorageRungeKuttaScheme lsrk_scheme = stage_5_order_4;
906 * Eventually, we select a detail of the spatial discretization, namely the
907 * numerical flux (Riemann solver) at the faces between cells. For
this
908 * program, we have implemented a modified variant of the Lax--Friedrichs
909 * flux and the Harten--Lax--van Leer (HLL) flux.
912 *
enum EulerNumericalFlux
914 * lax_friedrichs_modified,
915 * harten_lax_vanleer,
917 * constexpr EulerNumericalFlux numerical_flux_type = lax_friedrichs_modified;
924 * <a name=
"Equationdata"></a>
925 * <h3>Equation data</h3>
929 * We now define a
class with the exact solution for the test case 0 and
one
930 * with a background flow field
for test
case 1 of the channel. Given that
931 * the Euler equations are a problem with @f$d+2@f$ equations in @f$d@f$ dimensions,
932 * we need to tell the
Function base
class about the correct number of
937 *
class ExactSolution :
public Function<dim>
940 * ExactSolution(
const double time)
945 *
const unsigned int component = 0)
const override;
952 * As far as the actual
function implemented is concerned, the analytical
953 * test
case is an isentropic vortex
case (see
e.g. the book by Hesthaven
954 * and Warburton, Example 6.1 in Section 6.6 on page 209) which fulfills the
955 * Euler equations with
zero force term on the right hand side. Given that
956 * definition, we
return either the density, the momentum, or the energy
957 * depending on which component is requested. Note that the original
958 * definition of the density involves the @f$\frac{1}{
\gamma -1}@f$-th power of
959 * some expression. Since `
std::pow()` has pretty slow implementations on
960 * some systems, we replace it by logarithm followed by exponentiation (of
961 * base 2), which is mathematically equivalent but usually much better
962 * optimized. This formula might lose accuracy in the last digits
963 *
for very small
numbers compared to `
std::pow()`, but we are happy with
964 * it anyway, since small
numbers map to data close to 1.
968 * For the channel test
case, we simply select a density of 1, a velocity of
969 * 0.4 in @f$x@f$ direction and
zero in the other directions, and an energy that
970 * corresponds to a speed of sound of 1.3 measured against the background
971 * velocity field, computed from the relation @f$E = \frac{c^2}{
\gamma (\
gamma
972 * -1)} + \frac 12 \rho \|u\|^2@f$.
977 *
const unsigned int component)
const
979 *
const double t = this->
get_time();
986 *
const double beta = 5;
990 *
const double radius_sqr =
991 * (x - x0).norm_square() - 2. * (x[0] - x0[0]) * t + t * t;
992 *
const double factor =
993 * beta / (
numbers::PI * 2) * std::exp(1. - radius_sqr);
994 *
const double density_log = std::log2(
995 * std::abs(1. - (
gamma - 1.) /
gamma * 0.25 * factor * factor));
996 *
const double density = std::exp2(density_log * (1. / (
gamma - 1.)));
997 *
const double u = 1. - factor * (x[1] - x0[1]);
998 *
const double v = factor * (x[0] - t - x0[0]);
1000 *
if (component == 0)
1002 *
else if (component == 1)
1003 *
return density * u;
1004 *
else if (component == 2)
1005 *
return density * v;
1008 *
const double pressure =
1009 * std::exp2(density_log * (
gamma / (
gamma - 1.)));
1010 *
return pressure / (
gamma - 1.) +
1011 * 0.5 * (density * u * u + density * v * v);
1017 *
if (component == 0)
1019 *
else if (component == 1)
1021 *
else if (component == dim + 1)
1022 *
return 3.097857142857143;
1038 * <a name=
"LowstorageexplicitRungeKuttatimeintegrators"></a>
1039 * <h3>Low-storage
explicit Runge--Kutta time integrators</h3>
1043 * The next few lines implement a few low-storage variants of Runge--Kutta
1044 * methods. These methods have specific Butcher tableaux with coefficients
1045 * @f$b_i@f$ and @f$a_i@f$ as shown in the introduction. As usual in Runge--Kutta
1046 * method, we can deduce time steps, @f$c_i = \sum_{j=1}^{i-2} b_i + a_{i-1}@f$
1047 * from those coefficients. The main advantage of
this kind of scheme is the
1048 * fact that only two vectors are needed per stage, namely the accumulated
1049 * part of the solution @f$\mathbf{
w}@f$ (that will hold the solution
1050 * @f$\mathbf{
w}^{n+1}@f$ at the
new time @f$t^{n+1}@f$ after the last stage), the
1051 * update vector @f$\mathbf{r}_i@f$ that gets evaluated during the stages, plus
1052 *
one vector @f$\mathbf{k}_i@f$ to hold the evaluation of the
operator. Such a
1053 * Runge--Kutta setup reduces the memory storage and memory access. As the
1054 * memory bandwidth is often the performance-limiting factor on modern
1055 * hardware when the evaluation of the differential
operator is
1056 * well-optimized, performance can be improved over standard time
1057 * integrators. This is
true also when taking into account that a
1058 * conventional Runge--Kutta scheme might allow
for slightly larger time
1059 * steps as more
free parameters allow
for better stability properties.
1063 * In
this tutorial programs, we concentrate on a few variants of
1064 * low-storage schemes defined in the article by Kennedy, Carpenter, and
1065 * Lewis (2000), as well as
one variant described by Tselios and Simos
1066 * (2007). There is a large series of other schemes available, which could
1067 * be addressed by additional sets of coefficients or slightly different
1072 * We define a single
class for the four integrators, distinguished by the
1073 *
enum described above. To each scheme, we then fill the vectors
for the
1074 * @f$b_i@f$ and @f$a_i@f$ to the given variables in the
class.
1077 *
class LowStorageRungeKuttaIntegrator
1080 * LowStorageRungeKuttaIntegrator(
const LowStorageRungeKuttaScheme scheme)
1084 * First comes the three-stage scheme of order three by Kennedy et al.
1085 * (2000). While its stability region is significantly smaller than
for
1086 * the other schemes, it only involves three stages, so it is very
1087 * competitive in terms of the work per stage.
1092 *
case stage_3_order_3:
1094 * bi = {{0.245170287303492, 0.184896052186740, 0.569933660509768}};
1095 * ai = {{0.755726351946097, 0.386954477304099}};
1102 * The next scheme is a five-stage scheme of order four, again
1103 * defined in the paper by Kennedy et al. (2000).
1106 *
case stage_5_order_4:
1108 * bi = {{1153189308089. / 22510343858157.,
1109 * 1772645290293. / 4653164025191.,
1110 * -1672844663538. / 4480602732383.,
1111 * 2114624349019. / 3568978502595.,
1112 * 5198255086312. / 14908931495163.}};
1113 * ai = {{970286171893. / 4311952581923.,
1114 * 6584761158862. / 12103376702013.,
1115 * 2251764453980. / 15575788980749.,
1116 * 26877169314380. / 34165994151039.}};
1123 * The following scheme of seven stages and order four has been
1124 * explicitly derived
for acoustics problems. It is a balance of
1125 * accuracy
for imaginary
eigenvalues among fourth order schemes,
1126 * combined with a large stability region. Since DG schemes are
1127 * dissipative among the highest frequencies,
this does not
1128 * necessarily translate to the highest possible time step per
1129 * stage. In the context of the present tutorial program, the
1130 * numerical flux plays a crucial role in the dissipation and thus
1131 * also the maximal stable time step size. For the modified
1132 * Lax--Friedrichs flux,
this scheme is similar to the
1133 * `stage_5_order_4` scheme in terms of step size per stage
if only
1134 * stability is considered, but somewhat less efficient
for the HLL
1138 *
case stage_7_order_4:
1140 * bi = {{0.0941840925477795334,
1141 * 0.149683694803496998,
1142 * 0.285204742060440058,
1143 * -0.122201846148053668,
1144 * 0.0605151571191401122,
1145 * 0.345986987898399296,
1146 * 0.186627171718797670}};
1147 * ai = {{0.241566650129646868 + bi[0],
1148 * 0.0423866513027719953 + bi[1],
1149 * 0.215602732678803776 + bi[2],
1150 * 0.232328007537583987 + bi[3],
1151 * 0.256223412574146438 + bi[4],
1152 * 0.0978694102142697230 + bi[5]}};
1159 * The last scheme included here is the nine-stage scheme of order
1160 * five from Kennedy et al. (2000). It is the most accurate among
1161 * the schemes used here, but the higher order of accuracy
1162 * sacrifices some stability, so the step length normalized per
1163 * stage is less than
for the fourth order schemes.
1166 *
case stage_9_order_5:
1168 * bi = {{2274579626619. / 23610510767302.,
1169 * 693987741272. / 12394497460941.,
1170 * -347131529483. / 15096185902911.,
1171 * 1144057200723. / 32081666971178.,
1172 * 1562491064753. / 11797114684756.,
1173 * 13113619727965. / 44346030145118.,
1174 * 393957816125. / 7825732611452.,
1175 * 720647959663. / 6565743875477.,
1176 * 3559252274877. / 14424734981077.}};
1177 * ai = {{1107026461565. / 5417078080134.,
1178 * 38141181049399. / 41724347789894.,
1179 * 493273079041. / 11940823631197.,
1180 * 1851571280403. / 6147804934346.,
1181 * 11782306865191. / 62590030070788.,
1182 * 9452544825720. / 13648368537481.,
1183 * 4435885630781. / 26285702406235.,
1184 * 2357909744247. / 11371140753790.}};
1194 *
unsigned int n_stages() const
1201 * The main
function of the time integrator is to go through the stages,
1202 * evaluate the
operator, prepare the @f$\mathbf{r}_i@f$ vector
for the next
1203 * evaluation, and update the solution vector @f$\mathbf{
w}@f$. We hand off
1204 * the work to the `pde_operator` involved in order to be able to
merge
1205 * the vector operations of the Runge--Kutta setup with the evaluation of
1206 * the differential
operator for better performance, so all we
do here is
1207 * to delegate the vectors and coefficients.
1211 * We separately
call the
operator for the
first stage because we need
1212 * slightly modified arguments there: We evaluate the solution from
1213 * the old solution @f$\mathbf{
w}^n@f$ rather than a @f$\mathbf r_i@f$ vector, so
1214 * the
first argument is `solution`. We here let the stage vector
1215 * @f$\mathbf{r}_i@f$ also hold the temporary result of the evaluation, as it
1216 * is not used otherwise. For all subsequent stages, we use the vector
1217 * `vec_ki` as the
second vector argument to store the result of the
1218 *
operator evaluation. Finally, when we are at the last stage, we must
1219 * skip the computation of the vector @f$\mathbf{r}_{s+1}@f$ as there is no
1220 * coefficient @f$a_s@f$ available (nor will it be used).
1223 *
template <
typename VectorType,
typename Operator>
1224 *
void perform_time_step(
const Operator &pde_operator,
1225 *
const double current_time,
1226 *
const double time_step,
1233 * pde_operator.perform_stage(current_time,
1234 * bi[0] * time_step,
1235 * ai[0] * time_step,
1240 *
double sum_previous_bi = 0;
1241 *
for (
unsigned int stage = 1; stage < bi.size(); ++stage)
1243 *
const double c_i = sum_previous_bi + ai[stage - 1];
1244 * pde_operator.perform_stage(current_time + c_i * time_step,
1245 * bi[stage] * time_step,
1246 * (stage == bi.size() - 1 ?
1248 * ai[stage] * time_step),
1253 * sum_previous_bi += bi[stage - 1];
1258 * std::vector<double> bi;
1259 * std::vector<double> ai;
1267 * <a name=
"ImplementationofpointwiseoperationsoftheEulerequations"></a>
1268 * <h3>Implementation of
point-wise operations of the Euler equations</h3>
1272 * In the following
functions, we implement the various problem-specific
1273 * operators pertaining to the Euler equations. Each
function acts on the
1274 * vector of conserved variables @f$[\rho, \rho\mathbf{u},
E]@f$ that we hold in
1275 * the solution vectors, and computes various derived quantities.
1279 * First out is the computation of the velocity, that we derive from the
1280 * momentum variable @f$\rho \mathbf{u}@f$ by division by @f$\rho@f$. One thing to
1281 * note here is that we decorate all those
functions with the keyword
1283 * compiler-specific keyword that tells the compiler to never create a
1284 *
function call for any of those
functions, and instead move the
1286 * href=
"https://en.wikipedia.org/wiki/Inline_function">
inline</a> to where
1287 * they are called. This is critical
for performance because we
call into some
1288 * of those
functions millions or billions of times: For example, we both use
1289 * the velocity
for the computation of the flux further down, but also
for the
1290 * computation of the pressure, and both of these places are evaluated at
1291 * every quadrature
point of every cell. Making sure these
functions are
1292 * inlined ensures not only that the processor does not have to execute a jump
1293 * instruction into the
function (and the corresponding
return jump), but also
1294 * that the compiler can re-use intermediate information from
one function's
1295 * context in code that comes after the place where the function was called.
1296 * (We note that compilers are generally quite good at figuring out which
1297 * functions to inline by themselves. Here is a place where compilers may or
1298 * may not have figured it out by themselves but where we know for sure that
1299 * inlining is a win.)
1303 * Another trick we apply is a separate variable for the inverse density
1304 * @f$\frac{1}{\rho}@f$. This enables the compiler to only perform a single
1305 * division for the flux, despite the division being used at several
1306 * places. As divisions are around ten to twenty times as expensive as
1307 * multiplications or additions, avoiding redundant divisions is crucial for
1308 * performance. We note that taking the inverse first and later multiplying
1309 * with it is not equivalent to a division in floating point arithmetic due
1310 * to roundoff effects, so the compiler is not allowed to exchange one way by
1311 * the other with standard optimization flags. However, it is also not
1312 * particularly difficult to write the code in the right way.
1316 * To summarize, the chosen strategy of always inlining and careful
1317 * definition of expensive arithmetic operations allows us to write compact
1318 * code without passing all intermediate results around, despite making sure
1319 * that the code maps to excellent machine code.
1322 * template <int dim, typename Number>
1323 * inline DEAL_II_ALWAYS_INLINE
1324 * Tensor<1, dim, Number>
1325 * euler_velocity(const Tensor<1, dim + 2, Number> &conserved_variables)
1327 * const Number inverse_density = Number(1.) / conserved_variables[0];
1329 * Tensor<1, dim, Number> velocity;
1330 * for (unsigned int d = 0; d < dim; ++d)
1331 * velocity[d] = conserved_variables[1 + d] * inverse_density;
1338 * The next function computes the pressure from the vector of conserved
1339 * variables, using the formula @f$p = (\gamma - 1) \left(E - \frac 12 \rho
1340 * \mathbf{u}\cdot \mathbf{u}\right)@f$. As explained above, we use the
1341 * velocity from the `euler_velocity()` function. Note that we need to
1342 * specify the first template argument `dim` here because the compiler is
1343 * not able to deduce it from the arguments of the tensor, whereas the
1344 * second argument (number type) can be automatically deduced.
1347 * template <int dim, typename Number>
1348 * inline DEAL_II_ALWAYS_INLINE
1350 * euler_pressure(const Tensor<1, dim + 2, Number> &conserved_variables)
1352 * const Tensor<1, dim, Number> velocity =
1353 * euler_velocity<dim>(conserved_variables);
1355 * Number rho_u_dot_u = conserved_variables[1] * velocity[0];
1356 * for (unsigned int d = 1; d < dim; ++d)
1357 * rho_u_dot_u += conserved_variables[1 + d] * velocity[d];
1359 * return (gamma - 1.) * (conserved_variables[dim + 1] - 0.5 * rho_u_dot_u);
1364 * Here is the definition of the Euler flux function, i.e., the definition
1365 * of the actual equation. Given the velocity and pressure (that the
1366 * compiler optimization will make sure are done only once), this is
1367 * straight-forward given the equation stated in the introduction.
1370 * template <int dim, typename Number>
1371 * inline DEAL_II_ALWAYS_INLINE
1372 * Tensor<1, dim + 2, Tensor<1, dim, Number>>
1373 * euler_flux(const Tensor<1, dim + 2, Number> &conserved_variables)
1375 * const Tensor<1, dim, Number> velocity =
1376 * euler_velocity<dim>(conserved_variables);
1377 * const Number pressure = euler_pressure<dim>(conserved_variables);
1379 * Tensor<1, dim + 2, Tensor<1, dim, Number>> flux;
1380 * for (unsigned int d = 0; d < dim; ++d)
1382 * flux[0][d] = conserved_variables[1 + d];
1383 * for (unsigned int e = 0; e < dim; ++e)
1384 * flux[e + 1][d] = conserved_variables[e + 1] * velocity[d];
1385 * flux[d + 1][d] += pressure;
1386 * flux[dim + 1][d] =
1387 * velocity[d] * (conserved_variables[dim + 1] + pressure);
1395 * This next function is a helper to simplify the implementation of the
1396 * numerical flux, implementing the action of a tensor of tensors (with
1397 * non-standard outer dimension of size `dim + 2`, so the standard overloads
1398 * provided by deal.II's tensor classes
do not
apply here) with another
1399 * tensor of the same inner dimension, i.e., a
matrix-vector product.
1402 *
template <
int n_components,
int dim,
typename Number>
1416 * This
function implements the numerical flux (Riemann solver). It gets the
1417 * state from the two sides of an
interface and the normal vector, oriented
1418 * from the side of the solution @f$\mathbf{
w}^-@f$ towards the solution
1419 * @f$\mathbf{
w}^+@f$. In finite
volume methods which rely on piece-wise
1420 * constant data, the numerical flux is the central ingredient as it is the
1421 * only place where the physical information is entered. In DG methods, the
1422 * numerical flux is less central due to the polynomials within the elements
1423 * and the physical flux used there. As a result of higher-degree
1424 * interpolation with consistent values from both sides in the limit of a
1425 * continuous solution, the numerical flux can be seen as a control of the
1426 * jump of the solution from both sides to weakly impose continuity. It is
1427 * important to realize that a numerical flux alone cannot stabilize a
1428 * high-order DG method in the presence of shocks, and thus any DG method
1429 * must be combined with further shock-capturing techniques to handle those
1430 * cases. In
this tutorial, we focus on wave-like solutions of the Euler
1431 * equations in the subsonic regime without strong discontinuities where our
1432 * basic scheme is sufficient.
1436 * Nonetheless, the numerical flux is decisive in terms of the numerical
1437 * dissipation of the overall scheme and influences the admissible time step
1438 * size with
explicit Runge--Kutta methods. We consider two choices, a
1439 * modified Lax--Friedrichs scheme and the widely used Harten--Lax--van Leer
1440 * (HLL) flux. For both variants, we
first need to get the velocities and
1441 * pressures from both sides of the interface and evaluate the physical
1446 * For the local Lax--Friedrichs flux, the definition is @f$\hat{\mathbf{
F}}
1447 * =\frac{\mathbf{
F}(\mathbf{
w}^-)+\mathbf{
F}(\mathbf{
w}^+)}{2} +
1448 * \frac{\lambda}{2}\left[\mathbf{
w}^--\mathbf{
w}^+\right]\otimes
1449 * \mathbf{n^-}@f$, where the factor @f$\lambda =
1450 * \max\left(\|\mathbf{u}^-\|+c^-, \|\mathbf{u}^+\|+c^+\right)@f$ gives the
1451 * maximal wave speed and @f$c = \
sqrt{\lambda p / \rho}@f$ is the speed of
1452 * sound. Here, we choose two modifications of that expression
for reasons
1453 * of computational efficiency, given the small impact of the flux on the
1454 * solution. For the above definition of the factor @f$\lambda@f$, we would need
1455 * to take four square roots, two
for the two velocity norms and two
for the
1456 * speed of sound on either side. The
first modification is hence to rather
1457 * use @f$
\sqrt{\|\mathbf{u}\|^2+c^2}@f$ as an estimate of the maximal speed
1458 * (which is at most a factor of 2 away from the actual maximum, as shown in
1459 * the introduction). This allows us to pull the square root out of the
1460 * maximum and get away with a single square root computation. The
second
1461 * modification is to further relax on the parameter @f$\lambda@f$---the smaller
1462 * it is, the smaller the dissipation factor (which is multiplied by the
1463 * jump in @f$\mathbf{
w}@f$, which might result in a smaller or bigger
1464 * dissipation in the
end). This allows us to fit the spectrum into the
1465 * stability region of the
explicit Runge--Kutta integrator with bigger time
1466 * steps. However, we cannot make dissipation too small because otherwise
1467 * imaginary
eigenvalues grow larger. Finally, the current conservative
1468 * formulation is not energy-stable in the limit of @f$\lambda\to 0@f$ as it is
1469 * not skew-
symmetric, and would need additional measures such as
split-form
1470 * DG schemes in that
case.
1474 * For the HLL flux, we follow the formula from literature, introducing an
1475 * additional weighting of the two states from Lax--Friedrichs by a
1476 * parameter @f$s@f$. It is derived from the physical transport directions of
1477 * the Euler equations in terms of the current direction of velocity and
1478 * sound speed. For the velocity, we here choose a simple arithmetic average
1479 * which is sufficient
for DG scenarios and moderate jumps in material
1484 * Since the numerical flux is multiplied by the normal vector in the weak
1485 * form, we multiply by the result by the normal vector
for all terms in the
1486 * equation. In these multiplications, the `
operator*` defined above enables
1487 * a compact notation similar to the mathematical definition.
1491 * In
this and the following
functions, we use variable suffixes `_m` and
1492 * `_p` to indicate quantities derived from @f$\mathbf{
w}^-@f$ and @f$\mathbf{
w}^+@f$,
1493 * i.e., values
"here" and
"there" relative to the current cell when looking
1494 * at a neighbor cell.
1497 *
template <
int dim,
typename Number>
1504 *
const auto velocity_m = euler_velocity<dim>(u_m);
1505 *
const auto velocity_p = euler_velocity<dim>(u_p);
1507 *
const auto pressure_m = euler_pressure<dim>(u_m);
1508 *
const auto pressure_p = euler_pressure<dim>(u_p);
1510 *
const auto flux_m = euler_flux<dim>(u_m);
1511 *
const auto flux_p = euler_flux<dim>(u_p);
1513 *
switch (numerical_flux_type)
1515 *
case lax_friedrichs_modified:
1519 *
gamma * pressure_p * (1. / u_p[0]),
1520 * velocity_m.norm_square() +
1521 *
gamma * pressure_m * (1. / u_m[0])));
1523 *
return 0.5 * (flux_m * normal + flux_p * normal) +
1524 * 0.5 * lambda * (u_m - u_p);
1527 *
case harten_lax_vanleer:
1529 *
const auto avg_velocity_normal =
1530 * 0.5 * ((velocity_m + velocity_p) * normal);
1531 *
const auto avg_c =
std::sqrt(std::abs(
1533 * (pressure_p * (1. / u_p[0]) + pressure_m * (1. / u_m[0]))));
1534 *
const Number s_pos =
1535 *
std::max(Number(), avg_velocity_normal + avg_c);
1536 *
const Number s_neg =
1537 *
std::min(Number(), avg_velocity_normal - avg_c);
1538 *
const Number inverse_s = Number(1.) / (s_pos - s_neg);
1540 *
return inverse_s *
1541 * ((s_pos * (flux_m * normal) - s_neg * (flux_p * normal)) -
1542 * s_pos * s_neg * (u_m - u_p));
1557 * This and the next
function are helper
functions to provide compact
1558 * evaluation calls as multiple points get batched together via a
1559 *
VectorizedArray argument (see the @ref step_37
"step-37" tutorial
for details). This
1560 *
function is used
for the subsonic outflow boundary conditions where we
1561 * need to
set the energy component to a prescribed
value. The next
one
1562 * requests the solution on all components and is used
for inflow boundaries
1563 * where all components of the solution are
set.
1566 *
template <
int dim,
typename Number>
1570 *
const unsigned int component)
1573 *
for (
unsigned int v = 0; v < VectorizedArray<Number>::size(); ++v)
1576 *
for (
unsigned int d = 0;
d < dim; ++
d)
1577 * p[
d] = p_vectorized[
d][v];
1578 * result[v] =
function.value(p, component);
1584 *
template <
int dim,
typename Number,
int n_components = dim + 2>
1591 *
for (
unsigned int v = 0; v < VectorizedArray<Number>::size(); ++v)
1594 *
for (
unsigned int d = 0;
d < dim; ++
d)
1595 * p[
d] = p_vectorized[
d][v];
1597 * result[
d][v] =
function.
value(p,
d);
1607 * <a name=
"TheEulerOperationclass"></a>
1608 * <h3>The EulerOperation
class</h3>
1612 * This
class implements the evaluators for the Euler problem, in analogy to
1613 * the `LaplaceOperator`
class of @ref step_37 "step-37" or @ref step_59 "step-59". Since the present
1614 *
operator is non-linear and does not require a
matrix interface (to be
1615 * handed over to preconditioners), we skip the various `vmult`
functions
1616 * otherwise present in
matrix-
free operators and only implement an `
apply`
1617 *
function as well as the combination of `
apply` with the required vector
1618 * updates
for the low-storage Runge--Kutta time integrator mentioned above
1619 * (called `perform_stage`). Furthermore, we have added three additional
1621 * estimate of the time step scaling (that is combined with the Courant
1622 * number
for the actual time step size) based on the velocity and speed of
1623 * sound in the elements,
one for the projection of solutions (specializing
1625 * against a possible analytical solution or norms against some background
1630 * The rest of the
class is similar to other
matrix-
free tutorials. As
1631 * discussed in the introduction, we provide a few
functions to allow a user
1632 * to pass in various forms of boundary conditions on different parts of the
1634 * possible body forces.
1637 *
template <
int dim,
int degree,
int n_po
ints_1d>
1638 *
class EulerOperator
1641 *
static constexpr
unsigned int n_quadrature_points_1d = n_points_1d;
1651 *
void set_subsonic_outflow_boundary(
1657 *
void set_body_force(std::unique_ptr<
Function<dim>> body_force);
1659 *
void apply(
const double current_time,
1664 * perform_stage(
const Number cur_time,
1665 *
const Number factor_solution,
1666 *
const Number factor_ai,
1675 * std::array<double, 3> compute_errors(
1679 *
double compute_cell_transport_speed(
1690 * std::map<types::boundary_id, std::unique_ptr<Function<dim>>>
1691 * inflow_boundaries;
1692 * std::map<types::boundary_id, std::unique_ptr<Function<dim>>>
1693 * subsonic_outflow_boundaries;
1694 * std::set<types::boundary_id> wall_boundaries;
1695 * std::unique_ptr<Function<dim>> body_force;
1697 *
void local_apply_inverse_mass_matrix(
1701 *
const std::pair<unsigned int, unsigned int> & cell_range)
const;
1703 *
void local_apply_cell(
1707 *
const std::pair<unsigned int, unsigned int> & cell_range)
const;
1709 *
void local_apply_face(
1713 *
const std::pair<unsigned int, unsigned int> & cell_range)
const;
1715 *
void local_apply_boundary_face(
1719 *
const std::pair<unsigned int, unsigned int> & cell_range)
const;
1724 *
template <
int dim,
int degree,
int n_po
ints_1d>
1725 * EulerOperator<dim, degree, n_points_1d>::EulerOperator(
TimerOutput &timer)
1733 * For the initialization of the Euler
operator, we
set up the
MatrixFree
1734 * variable contained in the
class. This can be done given a mapping to
1735 * describe possible curved boundaries as well as a
DoFHandler object
1736 * describing the degrees of freedom. Since we use a discontinuous Galerkin
1737 * discretization in
this tutorial program where no constraints are imposed
1738 * strongly on the solution field, we
do not need to pass in an
1740 * construction. With respect to quadrature, we want to select two different
1741 * ways of computing the underlying integrals: The
first is a flexible
one,
1742 * based on a
template parameter `n_points_1d` (that will be assigned the
1743 * `n_q_points_1d`
value specified at the top of
this file). More accurate
1744 * integration is necessary to avoid the aliasing problem due to the
1745 * variable coefficients in the Euler
operator. The
second less accurate
1746 * quadrature formula is a tight
one based on `fe_degree+1` and needed
for
1747 * the inverse mass
matrix. While that formula provides an exact inverse
1748 * only on
affine element shapes and not on deformed elements, it enables
1749 * the fast inversion of the mass
matrix by tensor product techniques,
1750 * necessary to ensure optimal computational efficiency overall.
1753 * template <int dim, int degree, int n_points_1d>
1758 *
const std::vector<const DoFHandler<dim> *> dof_handlers = {&dof_handler};
1760 *
const std::vector<const AffineConstraints<double> *> constraints = {&
dummy};
1761 *
const std::vector<Quadrature<1>> quadratures = {
QGauss<1>(n_q_points_1d),
1778 * mapping, dof_handlers, constraints, quadratures, additional_data);
1783 *
template <
int dim,
int degree,
int n_po
ints_1d>
1784 *
void EulerOperator<dim, degree, n_points_1d>::initialize_vector(
1787 * data.initialize_dof_vector(vector);
1794 * The subsequent four member
functions are the ones that must be called from
1795 * outside to specify the various
types of boundaries. For an inflow boundary,
1796 * we must specify all components in terms of density @f$\rho@f$, momentum @f$\rho
1797 * \mathbf{u}@f$ and energy @f$E@f$. Given
this information, we then store the
1798 *
function alongside the respective boundary
id in a map member variable of
1799 *
this class. Likewise, we proceed
for the subsonic outflow boundaries (where
1800 * we request a
function as well, which we use to retrieve the energy) and
for
1801 * wall (no-penetration) boundaries where we impose
zero normal velocity (no
1802 *
function necessary, so we only request the boundary
id). For the present
1803 * DG code where boundary conditions are solely applied as part of the weak
1804 * form (during time integration), the
call to
set the boundary conditions
1805 * can appear both before or after the `
reinit()`
call to
this class. This
1806 * is different from continuous finite element codes where the boundary
1808 * sent into
MatrixFree for initialization, thus requiring to be
set before
1809 * the initialization of the
matrix-
free data structures.
1813 * The checks added in each of the four
function are used to
1814 * ensure that boundary conditions are mutually exclusive on the various
1815 * parts of the boundary, i.e., that a user does not accidentally designate a
1816 * boundary as both an inflow and say a subsonic outflow boundary.
1819 *
template <
int dim,
int degree,
int n_po
ints_1d>
1820 *
void EulerOperator<dim, degree, n_points_1d>::set_inflow_boundary(
1825 * subsonic_outflow_boundaries.end() &&
1826 * wall_boundaries.find(
boundary_id) == wall_boundaries.end(),
1827 *
ExcMessage(
"You already set the boundary with id " +
1829 *
" to another type of boundary before now setting " +
1831 *
AssertThrow(inflow_function->n_components == dim + 2,
1832 *
ExcMessage(
"Expected function with dim+2 components"));
1834 * inflow_boundaries[
boundary_id] = std::move(inflow_function);
1838 *
template <
int dim,
int degree,
int n_po
ints_1d>
1839 *
void EulerOperator<dim, degree, n_points_1d>::set_subsonic_outflow_boundary(
1844 * inflow_boundaries.end() &&
1845 * wall_boundaries.find(
boundary_id) == wall_boundaries.end(),
1846 *
ExcMessage(
"You already set the boundary with id " +
1848 *
" to another type of boundary before now setting " +
1849 *
"it as subsonic outflow"));
1850 *
AssertThrow(outflow_function->n_components == dim + 2,
1851 *
ExcMessage(
"Expected function with dim+2 components"));
1853 * subsonic_outflow_boundaries[
boundary_id] = std::move(outflow_function);
1857 *
template <
int dim,
int degree,
int n_po
ints_1d>
1858 *
void EulerOperator<dim, degree, n_points_1d>::set_wall_boundary(
1862 * inflow_boundaries.end() &&
1863 * subsonic_outflow_boundaries.find(
boundary_id) ==
1864 * subsonic_outflow_boundaries.end(),
1865 *
ExcMessage(
"You already set the boundary with id " +
1867 *
" to another type of boundary before now setting " +
1868 *
"it as wall boundary"));
1874 *
template <
int dim,
int degree,
int n_po
ints_1d>
1875 *
void EulerOperator<dim, degree, n_points_1d>::set_body_force(
1880 * this->body_force = std::move(body_force);
1888 * <a name=
"Localevaluators"></a>
1889 * <h4>Local evaluators</h4>
1893 * Now we proceed to the local evaluators
for the Euler problem. The
1894 * evaluators are relatively simple and follow what has been presented in
1895 * @ref step_37
"step-37", @ref step_48
"step-48", or @ref step_59
"step-59". The
first notable difference is the fact
1896 * that we use an
FEEvaluation with a non-standard number of quadrature
1897 * points. Whereas we previously
always set the number of quadrature points
1898 * to
equal the polynomial degree plus
one (ensuring exact integration on
1899 *
affine element shapes), we now
set the number quadrature points as a
1900 * separate variable (
e.g. the polynomial degree plus two or three halves of
1901 * the polynomial degree) to more accurately handle nonlinear terms. Since
1902 * the evaluator is fed with the appropriate
loop lengths via the
template
1903 * argument and keeps the number of quadrature points in the whole cell in
1905 * the more accurate formula without further changes.
1909 * The
second difference is due to the fact that we are now evaluating a
1910 * multi-component system, as opposed to the scalar systems considered
1911 * previously. The
matrix-
free framework provides several ways to handle the
1912 * multi-component
case. The variant shown here utilizes an
FEEvaluation
1913 *
object with multiple components embedded into it, specified by the fourth
1914 *
template argument `dim + 2`
for the components in the Euler system. As a
1915 * consequence, the
return type of FEEvaluation::get_value() is not a scalar
1916 * any more (that would return a
VectorizedArray type, collecting data from
1917 * several elements), but a
Tensor of `dim+2` components. The functionality
1918 * is otherwise similar to the scalar case; it is handled by a template
1920 * variant would have been to use several
FEEvaluation objects, a scalar
one
1921 * for the density, a vector-valued
one with `dim` components for the
1922 * momentum, and another scalar evaluator for the energy. To ensure that
1923 * those components
point to the correct part of the solution, the
1924 * constructor of
FEEvaluation takes three optional integer arguments after
1926 * multi-
DoFHandler systems (taking the
first by default), the number of the
1927 * quadrature
point in case there are multiple
Quadrature objects (see more
1928 * below), and as a third argument the component within a vector system. As
1929 * we have a single vector for all components, we would go with the third
1930 * argument, and
set it to `0` for the density, `1` for the vector-valued
1931 * momentum, and `dim+1` for the energy slot.
FEEvaluation then picks the
1932 * appropriate subrange of the solution vector during
1934 *
FEEvaluation::distributed_local_to_global() or the more compact
1940 * When it comes to the evaluation of the body force vector, we distinguish
1941 * between two cases for efficiency reasons: In case we have a constant
1943 * the
value outside the
loop over quadrature points and simply use the
1945 * the `evaluate_function()` method we provided above; this path is more
1946 * expensive because we need to access the memory associated with the
1947 * quadrature
point data.
1951 * The rest follows the other tutorial programs. Since we have implemented
1952 * all physics for the Euler equations in the separate `euler_flux()`
1953 * function, all we have to do here is to
call this function
1954 * given the current solution evaluated at quadrature points, returned by
1955 * `phi.get_value(q)`, and tell the
FEEvaluation object to queue the flux
1956 * for testing it by the gradients of the shape
functions (which is a
Tensor
1957 * of outer `dim+2` components, each holding a tensor of `dim` components
1958 * for the @f$x,y,z@f$ component of the Euler flux). One final thing worth
1959 * mentioning is the order in which we queue the data for testing by the
1960 *
value of the test function, `phi.submit_value()`, in case we are given an
1961 * external function: We must do this after calling `phi.get_value(q)`,
1962 * because `get_value()` (reading the solution) and `submit_value()`
1963 * (queuing the
value for multiplication by the test function and summation
1964 * over quadrature points) access the same underlying data field. Here it
1965 * would be easy to achieve also without temporary variable `w_q` since
1966 * there is no mixing between values and gradients. For more complicated
1968 * quadrature
point and then queue results again by
1975 * The interfaces imposes the present list of arguments, but since we are in
1976 * a member function where the
MatrixFree object is already available as the
1977 * `data` variable, we stick with that to avoid confusion.
1980 * template <
int dim,
int degree,
int n_points_1d>
1981 *
void EulerOperator<dim, degree, n_points_1d>::local_apply_cell(
1985 * const std::pair<
unsigned int,
unsigned int> & cell_range) const
1993 *
if (constant_function)
1994 * constant_body_force = evaluate_function<dim, Number, dim>(
1997 *
for (
unsigned int cell = cell_range.first; cell < cell_range.second; ++cell)
2000 * phi.gather_evaluate(src,
true,
false);
2002 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2004 *
const auto w_q = phi.get_value(q);
2005 * phi.submit_gradient(euler_flux<dim>(w_q), q);
2006 *
if (body_force.get() !=
nullptr)
2009 * constant_function ? constant_body_force :
2010 * evaluate_function<dim, Number, dim>(
2011 * *body_force, phi.quadrature_point(q));
2014 *
for (
unsigned int d = 0;
d < dim; ++
d)
2015 * forcing[
d + 1] = w_q[0] * force[
d];
2016 *
for (
unsigned int d = 0;
d < dim; ++
d)
2017 * forcing[dim + 1] += force[
d] * w_q[
d + 1];
2019 * phi.submit_value(forcing, q);
2023 * phi.integrate_scatter(body_force.get() !=
nullptr,
true, dst);
2031 * The next
function concerns the computation of integrals on interior
2032 * faces, where we need evaluators from both cells adjacent to the face. We
2033 * associate the variable `phi_m` with the solution component @f$\mathbf{
w}^-@f$
2034 * and the variable `phi_p` with the solution component @f$\mathbf{
w}^+@f$. We
2036 *
second argument, with `
true`
for the interior side and `
false`
for the
2037 * exterior side, with interior and exterior denoting the orientation with
2038 * respect to the normal vector.
2044 * and the
sum factorization parts. This combined operation not only saves a
2045 * line of code, but also contains an important optimization: Given that we
2046 * use a nodal basis in terms of the Lagrange polynomials in the points of
2047 * the Gauss-Lobatto quadrature formula, only @f$(p+1)^{
d-1}@f$ out of the
2048 * @f$(p+1)^
d@f$ basis
functions evaluate to non-
zero on each face. Thus, the
2049 * evaluator only accesses the necessary data in the vector and skips the
2050 * parts which are multiplied by
zero. If we had
first read the vector, we
2051 * would have needed to load all data from the vector, as the
call in
2052 * isolation would not know what data is required in subsequent
2054 * values and derivatives, indeed all @f$(p+1)^
d@f$ vector entries
for each
2055 * component are needed, as the normal derivative is
nonzero for all basis
2060 * The arguments to the evaluators as well as the procedure is similar to
2061 * the cell evaluation. We again use the more accurate (over-)integration
2062 * scheme due to the nonlinear terms, specified as the third
template
2063 * argument in the list. At the quadrature points, we then go to our
2064 *
free-standing
function for the numerical flux. It receives the solution
2065 * evaluated at quadrature points from both sides (i.e., @f$\mathbf{w}^-@f$ and
2066 * @f$\mathbf{w}^+@f$), as well as the normal vector onto the minus side. As
2067 * explained above, the numerical flux is already multiplied by the normal
2068 * vector from the minus side. We need to
switch the
sign because the
2069 * boundary term comes with a minus
sign in the weak form derived in the
2070 * introduction. The flux is then queued
for testing both on the minus
sign
2071 * and on the plus
sign, with switched
sign as the normal vector from the
2072 * plus side is exactly opposed to the
one from the minus side.
2075 *
template <
int dim,
int degree,
int n_po
ints_1d>
2076 *
void EulerOperator<dim, degree, n_points_1d>::local_apply_face(
2080 *
const std::pair<unsigned int, unsigned int> & face_range)
const
2087 *
for (
unsigned int face = face_range.first; face < face_range.second; ++face)
2089 * phi_p.reinit(face);
2090 * phi_p.gather_evaluate(src,
true,
false);
2092 * phi_m.reinit(face);
2093 * phi_m.gather_evaluate(src,
true,
false);
2095 *
for (
unsigned int q = 0; q < phi_m.n_q_points; ++q)
2097 *
const auto numerical_flux =
2098 * euler_numerical_flux<dim>(phi_m.get_value(q),
2099 * phi_p.get_value(q),
2100 * phi_m.get_normal_vector(q));
2101 * phi_m.submit_value(-numerical_flux, q);
2102 * phi_p.submit_value(numerical_flux, q);
2105 * phi_p.integrate_scatter(
true,
false, dst);
2106 * phi_m.integrate_scatter(
true,
false, dst);
2114 * For faces located at the boundary, we need to impose the appropriate
2115 * boundary conditions. In
this tutorial program, we implement four cases as
2116 * mentioned above. (
A fifth
case,
for supersonic outflow conditions is
2117 * discussed in the
"Results" section below.) The discontinuous Galerkin
2118 * method imposes boundary conditions not as constraints, but only
2119 * weakly. Thus, the various conditions are imposed by finding an appropriate
2120 * <i>exterior</i> quantity @f$\mathbf{
w}^+@f$ that is then handed to the
2121 * numerical flux
function also used
for the interior faces. In essence,
2122 * we
"pretend" a state on the outside of the domain in such a way that
2123 *
if that were reality, the solution of the PDE would satisfy the boundary
2124 * conditions we want.
2128 * For wall boundaries, we need to impose a no-normal-flux condition on the
2129 * momentum variable, whereas we use a Neumann condition
for the density and
2130 * energy with @f$\rho^+ = \rho^-@f$ and @f$E^+ =
E^-@f$. To achieve the no-normal
2131 * flux condition, we
set the exterior values to the interior values and
2132 * subtract two times the velocity in wall-normal direction, i.e., in the
2133 * direction of the normal vector.
2137 * For inflow boundaries, we simply
set the given Dirichlet data
2138 * @f$\mathbf{
w}_\mathrm{D}@f$ as a boundary
value. An alternative would have been
2139 * to use @f$\mathbf{
w}^+ = -\mathbf{
w}^- + 2 \mathbf{
w}_\mathrm{D}@f$, the
2140 * so-called mirror principle.
2144 * The imposition of outflow is essentially a Neumann condition, i.e.,
2145 * setting @f$\mathbf{
w}^+ = \mathbf{
w}^-@f$. For the
case of subsonic outflow,
2146 * we still need to impose a
value for the energy, which we derive from the
2147 * respective
function.
A special step is needed
for the
case of
2148 * <i>backflow</i>, i.e., the
case where there is a momentum flux into the
2149 * domain on the Neumann portion. According to the literature (a fact that can
2150 * be derived by appropriate energy arguments), we must
switch to another
2151 * variant of the flux on inflow parts, see Gravemeier, Comerford,
2152 * Yoshihara, Ismail, Wall,
"A novel formulation for Neumann inflow
2153 * conditions in biomechanics", Int. J. Numer. Meth. Biomed. Eng., vol. 28
2154 * (2012). Here, the momentum term needs to be added once again, which
2155 * corresponds to removing the flux contribution on the momentum
2156 * variables. We
do this in a post-processing step, and only
for the
case
2157 * when we both are at an outflow boundary and the dot product between the
2158 * normal vector and the momentum (or, equivalently, velocity) is
2159 * negative. As we work on data of several quadrature points at once
for
2160 * SIMD vectorizations, we here need to explicitly
loop over the array
2161 * entries of the SIMD array.
2165 * In the implementation below, we check
for the various
types
2166 * of boundaries at the
level of quadrature points. Of course, we could also
2167 * have moved the decision out of the quadrature
point loop and treat entire
2168 * faces as of the same kind, which avoids some map/
set lookups in the inner
2169 *
loop over quadrature points. However, the loss of efficiency is hardly
2170 * noticeable, so we opt
for the simpler code here. Also note that the
final
2171 * `
else` clause will
catch the
case when some part of the boundary was not
2172 * assigned any boundary condition via `EulerOperator::set_..._boundary(...)`.
2175 *
template <
int dim,
int degree,
int n_po
ints_1d>
2176 *
void EulerOperator<dim, degree, n_points_1d>::local_apply_boundary_face(
2180 *
const std::pair<unsigned int, unsigned int> & face_range)
const
2184 *
for (
unsigned int face = face_range.first; face < face_range.second; ++face)
2187 * phi.gather_evaluate(src,
true,
false);
2189 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2191 *
const auto w_m = phi.get_value(q);
2192 *
const auto normal = phi.get_normal_vector(q);
2194 *
auto rho_u_dot_n = w_m[1] * normal[0];
2195 *
for (
unsigned int d = 1;
d < dim; ++
d)
2196 * rho_u_dot_n += w_m[1 +
d] * normal[
d];
2198 *
bool at_outflow =
false;
2201 *
const auto boundary_id = data.get_boundary_id(face);
2202 *
if (wall_boundaries.find(
boundary_id) != wall_boundaries.end())
2205 *
for (
unsigned int d = 0;
d < dim; ++
d)
2206 * w_p[
d + 1] = w_m[
d + 1] - 2. * rho_u_dot_n * normal[
d];
2207 * w_p[dim + 1] = w_m[dim + 1];
2210 * inflow_boundaries.end())
2212 * evaluate_function(*inflow_boundaries.find(
boundary_id)->second,
2213 * phi.quadrature_point(q));
2214 *
else if (subsonic_outflow_boundaries.find(
boundary_id) !=
2215 * subsonic_outflow_boundaries.end())
2218 * w_p[dim + 1] = evaluate_function(
2219 * *subsonic_outflow_boundaries.find(
boundary_id)->second,
2220 * phi.quadrature_point(q),
2222 * at_outflow =
true;
2227 *
"you set a boundary condition for "
2228 *
"this part of the domain boundary?"));
2230 *
auto flux = euler_numerical_flux<dim>(w_m, w_p, normal);
2233 *
for (
unsigned int v = 0; v < VectorizedArray<Number>::size(); ++v)
2235 *
if (rho_u_dot_n[v] < -1
e-12)
2236 *
for (
unsigned int d = 0;
d < dim; ++
d)
2237 * flux[
d + 1][v] = 0.;
2240 * phi.submit_value(-flux, q);
2243 * phi.integrate_scatter(
true,
false, dst);
2251 * The next
function implements the inverse mass
matrix operation. The
2252 * algorithms and rationale have been discussed extensively in the
2253 * introduction, so we here limit ourselves to the technicalities of the
2255 * operations as the forward evaluation of the mass
matrix, except with a
2256 * different interpolation
matrix, representing the inverse @f$S^{-1}@f$
2257 * factors. These represent a change of basis from the specified basis (in
2258 *
this case, the Lagrange basis in the points of the Gauss--Lobatto
2259 * quadrature formula) to the Lagrange basis in the points of the Gauss
2260 * quadrature formula. In the latter basis, we can
apply the inverse of the
2261 *
point-wise `JxW` factor, i.e., the quadrature weight times the
2262 *
determinant of the Jacobian of the mapping from reference to real
2263 * coordinates. Once
this is done, the basis is changed back to the nodal
2264 * Gauss-Lobatto basis again. All of these operations are done by the
2265 * `
apply()`
function below. What we need to provide is the local fields to
2267 *
object) and write the results back to the destination vector of the mass
2272 * One thing to note is that we added two integer arguments (that are
2275 * only have
one) and the
second being 1 to make the quadrature formula
2276 * selection. As we use the quadrature formula 0
for the over-integration of
2277 * nonlinear terms, we use the formula 1 with the
default @f$p+1@f$ (or
2278 * `fe_degree+1` in terms of the variable name) points
for the mass
2279 *
matrix. This leads to square contributions to the mass
matrix and ensures
2280 * exact integration, as explained in the introduction.
2283 *
template <
int dim,
int degree,
int n_po
ints_1d>
2284 *
void EulerOperator<dim, degree, n_points_1d>::local_apply_inverse_mass_matrix(
2288 *
const std::pair<unsigned int, unsigned int> & cell_range)
const
2294 *
for (
unsigned int cell = cell_range.first; cell < cell_range.second; ++cell)
2297 * phi.read_dof_values(src);
2299 * inverse.apply(phi.begin_dof_values(), phi.begin_dof_values());
2301 * phi.set_dof_values(dst);
2310 * <a name=
"Theapplyandrelatedfunctions"></a>
2315 * We now come to the function which implements the evaluation of the Euler
2316 * operator as a whole, i.
e., @f$\mathcal M^{-1} \mathcal
L(t, \mathbf{
w})@f$,
2317 * calling into the local evaluators presented above. The steps should be
2318 * clear from the previous code. One thing to note is that we need to adjust
2319 * the time in the
functions we have associated with the various parts of
2320 * the boundary, in order to be consistent with the equation in
case the
2322 * perform the cell and face integrals, including the necessary ghost data
2323 * exchange in the `src` vector. The seventh argument to the
function,
2324 * `
true`, specifies that we want to
zero the `dst` vector as part of the
2325 *
loop, before we start accumulating integrals into it. This variant is
2326 * preferred over explicitly calling `dst = 0.;` before the
loop as the
2327 * zeroing operation is done on a subrange of the vector in parts that are
2328 * written by the integrals nearby. This enhances data locality and allows
2329 *
for caching, saving
one roundtrip of vector data to main memory and
2330 * enhancing performance. The last two arguments to the
loop determine which
2331 * data is exchanged: Since we only access the values of the shape
functions
2332 *
one faces, typical of
first-order hyperbolic problems, and since we have
2333 * a nodal basis with nodes at the reference element surface, we only need
2334 * to exchange those parts. This again saves precious memory bandwidth.
2338 * Once the spatial
operator @f$\mathcal
L@f$ is applied, we need to make a
2341 * is cheaper than the full
loop as access only goes to the degrees of
2342 * freedom associated with the locally owned cells, which is simply the
2343 * locally owned degrees of freedom for DG discretizations. Thus, no ghost
2344 * exchange is needed here.
2348 * Around all these
functions, we put timer scopes to record the
2349 * computational time for statistics about the contributions of the various
2353 * template <
int dim,
int degree,
int n_points_1d>
2354 *
void EulerOperator<dim, degree, n_points_1d>::
apply(
2355 * const
double current_time,
2362 *
for (
auto &i : inflow_boundaries)
2363 * i.second->set_time(current_time);
2364 *
for (
auto &i : subsonic_outflow_boundaries)
2365 * i.second->set_time(current_time);
2367 * data.loop(&EulerOperator::local_apply_cell,
2368 * &EulerOperator::local_apply_face,
2369 * &EulerOperator::local_apply_boundary_face,
2381 * data.cell_loop(&EulerOperator::local_apply_inverse_mass_matrix,
2392 * Let us move to the
function that does an entire stage of a Runge--Kutta
2394 * to the vectors, namely `next_ri = solution + factor_ai * k_i` and
2395 * `solution += factor_solution * k_i`. Rather than performing these
2396 * steps through the vector interfaces, we here present an alternative
2397 * strategy that is faster on cache-based architectures. As the memory
2398 * consumed by the vectors is often much larger than what fits into caches,
2399 * the data has to effectively come from the slow RAM memory. The situation
2400 * can be improved by
loop fusion, i.
e., performing both the updates to
2401 * `next_ki` and `solution` within a single sweep. In that case, we would
2402 * read the two vectors `rhs` and `solution` and write into `next_ki` and
2403 * `solution`, compared to at least 4 reads and two writes in the baseline
2404 * case. Here, we go
one step further and perform the
loop immediately when
2405 * the mass
matrix inversion has finished on a part of the
2406 * vector.
MatrixFree::cell_loop() provides a mechanism to attach an
2407 * `std::function` both before the
loop over cells
first touches a vector
2408 * entry (which we do not use here, but is
e.g. used for zeroing the vector)
2409 * and a
second `std::function` to be called after the
loop last touches
2410 * an entry. The callback is in form of a range over the given vector (in
2411 * terms of the local index numbering in the MPI universe) that can be
2412 * addressed by `local_element()`
functions. For this
second callback, we
2413 * create a lambda that works on a range and write the respective update on
2415 * to suggest to the compiler to SIMD parallelize this
loop (which means in
2416 * practice that we ensure that there is no overlap, also called
2417 * aliasing, between the index ranges of the pointers we use inside the
2418 * loops). Note that we select a different code path for the last
2419 * Runge--Kutta stage when we do not need to update the `next_ri`
2420 * vector. This strategy gives a considerable speedup. Whereas the inverse
2421 * mass
matrix and vector updates take more than 60% of the computational
2422 * time with default vector updates on a 40-core machine, the percentage is
2423 * around 35% with the more optimized variant. In other words, this is a
2424 * speedup of around a third.
2427 * template <
int dim,
int degree,
int n_points_1d>
2428 *
void EulerOperator<dim, degree, n_points_1d>::perform_stage(
2429 * const Number current_time,
2430 * const Number factor_solution,
2431 * const Number factor_ai,
2440 *
for (
auto &i : inflow_boundaries)
2441 * i.second->set_time(current_time);
2442 *
for (
auto &i : subsonic_outflow_boundaries)
2443 * i.second->set_time(current_time);
2445 * data.loop(&EulerOperator::local_apply_cell,
2446 * &EulerOperator::local_apply_face,
2447 * &EulerOperator::local_apply_boundary_face,
2460 * &EulerOperator::local_apply_inverse_mass_matrix,
2464 * std::function<
void(
const unsigned int,
const unsigned int)>(),
2465 * [&](
const unsigned int start_range,
const unsigned int end_range) {
2466 *
const Number ai = factor_ai;
2467 *
const Number bi = factor_solution;
2468 *
if (ai == Number())
2471 *
for (
unsigned int i = start_range; i < end_range; ++i)
2473 *
const Number k_i = next_ri.local_element(i);
2474 *
const Number sol_i = solution.local_element(i);
2475 * solution.local_element(i) = sol_i + bi * k_i;
2481 *
for (
unsigned int i = start_range; i < end_range; ++i)
2483 *
const Number k_i = next_ri.local_element(i);
2484 *
const Number sol_i = solution.local_element(i);
2485 * solution.local_element(i) = sol_i + bi * k_i;
2486 * next_ri.local_element(i) = sol_i + ai * k_i;
2497 * Having discussed the implementation of the
functions that deal with
2498 * advancing the solution by
one time step, let us now move to
functions
2499 * that implement other, ancillary operations. Specifically, these are
2500 *
functions that compute projections, evaluate errors, and compute the speed
2501 * of information transport on a cell.
2507 * elements where there is no need to
set up and solve a linear system, as
2508 * each element has independent basis
functions. The reason why we show the
2509 * code here, besides a small speedup of
this non-critical operation, is that
2510 * it shows additional functionality provided by
2515 * The projection operation works as follows: If we denote the
matrix of
2516 * shape
functions evaluated at quadrature points by @f$S@f$, the projection on
2517 * cell @f$K@f$ is an operation of the form @f$\underbrace{S J^K S^\mathrm
2518 *
T}_{\mathcal M^K} \mathbf{
w}^K = S J^K
2519 * \tilde{\mathbf{
w}}(\mathbf{x}_q)_{q=1:n_q}@f$, where @f$J^K@f$ is the
diagonal
2521 * weight (JxW), @f$\mathcal M^K@f$ is the cell-wise mass
matrix, and
2522 * @f$\tilde{\mathbf{
w}}(\mathbf{x}_q)_{q=1:n_q}@f$ is the evaluation of the
2523 * field to be projected onto quadrature points. (In reality the
matrix @f$S@f$
2524 * has additional structure through the tensor product, as explained in the
2525 * introduction.) This system can now equivalently be written as
2526 * @f$\mathbf{
w}^K = \left(S J^K S^\mathrm
T\right)^{-1} S J^K
2527 * \tilde{\mathbf{
w}}(\mathbf{x}_q)_{q=1:n_q} = S^{-\mathrm
T}
2528 * \left(J^K\right)^{-1} S^{-1} S J^K
2529 * \tilde{\mathbf{
w}}(\mathbf{x}_q)_{q=1:n_q}@f$. Now, the term @f$S^{-1} S@f$ and
2530 * then @f$\left(J^K\right)^{-1} J^K@f$ cancel, resulting in the
final
2531 * expression @f$\mathbf{
w}^K = S^{-\mathrm
T}
2532 * \tilde{\mathbf{
w}}(\mathbf{x}_q)_{q=1:n_q}@f$. This operation is
2535 * The name is derived from the fact that
this projection is simply
2536 * the multiplication by @f$S^{-\mathrm
T}@f$, a basis change from the
2537 * nodal basis in the points of the Gaussian quadrature to the given finite
2538 * element basis. Note that we
call FEEvaluation::set_dof_values() to write
2539 * the result into the vector, overwriting previous content, rather than
2540 * accumulating the results as typical in integration tasks -- we can do
2541 * this because every vector entry has contributions from only a single
2542 * cell for discontinuous Galerkin discretizations.
2545 * template <
int dim,
int degree,
int n_points_1d>
2546 *
void EulerOperator<dim, degree, n_points_1d>::
project(
2553 * solution.zero_out_ghosts();
2554 *
for (
unsigned int cell = 0; cell < data.n_macro_cells(); ++cell)
2557 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2558 * phi.submit_dof_value(evaluate_function(
function,
2559 * phi.quadrature_point(q)),
2561 * inverse.transform_from_q_points_to_basis(dim + 2,
2562 * phi.begin_dof_values(),
2563 * phi.begin_dof_values());
2564 * phi.set_dof_values(solution);
2572 * The next
function again repeats functionality also provided by the
2574 * the
explicit code to highlight how the vectorization across several cells
2575 * works and how to accumulate results via that interface: Recall that each
2576 * <i>lane</i> of the vectorized array holds data from a different cell. By
2577 * the
loop over all cell batches that are owned by the current MPI process,
2579 * we would need to further go on and
sum across the entries in the SIMD
2580 * array. However, such a procedure is not stable as the SIMD array could in
2581 * fact not hold
valid data
for all its lanes. This happens when the number
2582 * of locally owned cells is not a multiple of the SIMD width. To avoid
2583 *
invalid data, we must explicitly skip those
invalid lanes when accessing
2584 * the data. While
one could imagine that we could make it work by simply
2585 * setting the empty lanes to
zero (and thus, not contribute to a
sum), the
2586 * situation is more complicated than that: What
if we were to compute a
2587 * velocity out of the momentum? Then, we would need to divide by the
2588 * density, which is
zero -- the result would consequently be NaN and
2589 * contaminate the result. This trap is avoided by accumulating the results
2590 * from the
valid SIMD range as we
loop through the cell batches,
using the
2593 * most cells, but can be less on the last cell batch if the number of cells
2594 * has a remainder compared to the SIMD width.
2597 * template <
int dim,
int degree,
int n_points_1d>
2598 * std::array<
double, 3> EulerOperator<dim, degree, n_points_1d>::compute_errors(
2603 *
double errors_squared[3] = {};
2606 *
for (
unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
2609 * phi.gather_evaluate(solution,
true,
false);
2611 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2613 *
const auto error =
2614 * evaluate_function(
function, phi.quadrature_point(q)) -
2616 *
const auto JxW = phi.JxW(q);
2618 * local_errors_squared[0] += error[0] * error[0] * JxW;
2619 *
for (
unsigned int d = 0;
d < dim; ++
d)
2620 * local_errors_squared[1] += (error[
d + 1] * error[
d + 1]) * JxW;
2621 * local_errors_squared[2] += (error[dim + 1] * error[dim + 1]) * JxW;
2623 *
for (
unsigned int v = 0; v < data.n_active_entries_per_cell_batch(cell);
2625 *
for (
unsigned int d = 0;
d < 3; ++
d)
2626 * errors_squared[
d] += local_errors_squared[
d][v];
2631 * std::array<double, 3> errors;
2632 *
for (
unsigned int d = 0;
d < 3; ++
d)
2633 * errors[
d] = std::sqrt(errors_squared[
d]);
2642 * This
final function of the EulerOperator
class is used to estimate the
2643 * transport speed, scaled by the mesh size, that is relevant
for setting
2644 * the time step size in the
explicit time integrator. In the Euler
2645 * equations, there are two speeds of transport, namely the convective
2646 * velocity @f$\mathbf{u}@f$ and the propagation of sound waves with sound
2647 * speed @f$c =
\sqrt{
\gamma p/\rho}@f$ relative to the medium moving at
2648 * velocity @f$\mathbf u@f$.
2652 * In the formula
for the time step size, we are interested not by
2653 * these absolute speeds, but by the amount of time it takes
for
2654 * information to cross a single cell. For information transported along with
2655 * the medium, @f$\mathbf u@f$ is scaled by the mesh size,
2656 * so an estimate of the maximal velocity can be obtained by computing
2657 * @f$\|J^{-\mathrm
T} \mathbf{u}\|_\infty@f$, where @f$J@f$ is the Jacobian of the
2658 * transformation from real to the reference domain. Note that
2660 * Jacobian, representing the metric term from real to reference
2661 * coordinates, so we do not need to
transpose it again. We store this limit
2662 * in the variable `convective_limit` in the code below.
2666 * The sound propagation is isotropic, so we need to take mesh sizes in any
2667 * direction into account. The appropriate mesh size scaling is then given
2668 * by the minimal singular
value of @f$J@f$ or, equivalently, the maximal
2670 * by the minimal distance between
vertices of a cell when ignoring curved
2671 * cells. To get the maximal singular
value of the Jacobian, the
general
2672 * strategy would be some LAPACK
function. Since all we need here is an
2673 * estimate, we can avoid the hassle of decomposing a tensor of
2675 * eigenvalue
function without vectorization, and instead use a few
2676 * iterations (five in the code below) of the power method applied to
2677 * @f$J^{-1}J^{-\mathrm
T}@f$. The speed of convergence of
this method depends
2678 * on the ratio of the largest to the next largest eigenvalue and the
2679 *
initial guess, which is the vector of all ones. This might suggest that
2680 * we get slow convergence on cells close to a cube shape where all
2681 * lengths are almost the same. However,
this slow convergence means that
2682 * the result will sit between the two largest singular values, which both
2683 * are close to the maximal
value anyway. In all other cases, convergence
2684 * will be quick. Thus, we can merely hardcode 5 iterations here and be
2685 * confident that the result is good.
2688 *
template <
int dim,
int degree,
int n_po
ints_1d>
2689 *
double EulerOperator<dim, degree, n_points_1d>::compute_cell_transport_speed(
2693 * Number max_transport = 0;
2696 *
for (
unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
2699 * phi.gather_evaluate(solution,
true,
false);
2701 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2703 *
const auto solution = phi.get_value(q);
2704 *
const auto velocity = euler_velocity<dim>(solution);
2705 *
const auto pressure = euler_pressure<dim>(solution);
2707 *
const auto inverse_jacobian = phi.inverse_jacobian(q);
2708 *
const auto convective_speed = inverse_jacobian * velocity;
2710 *
for (
unsigned int d = 0;
d < dim; ++
d)
2711 * convective_limit =
2712 *
std::max(convective_limit, std::abs(convective_speed[
d]));
2714 *
const auto speed_of_sound =
2718 *
for (
unsigned int d = 0;
d < dim; ++
d)
2719 * eigenvector[
d] = 1.;
2720 *
for (
unsigned int i = 0; i < 5; ++i)
2722 * eigenvector =
transpose(inverse_jacobian) *
2723 * (inverse_jacobian * eigenvector);
2725 *
for (
unsigned int d = 0;
d < dim; ++
d)
2726 * eigenvector_norm =
2727 *
std::max(eigenvector_norm, std::abs(eigenvector[
d]));
2728 * eigenvector /= eigenvector_norm;
2730 *
const auto jac_times_ev = inverse_jacobian * eigenvector;
2731 *
const auto max_eigenvalue =
std::sqrt(
2732 * (jac_times_ev * jac_times_ev) / (eigenvector * eigenvector));
2735 * max_eigenvalue * speed_of_sound + convective_limit);
2740 * Similarly to the previous
function, we must make sure to accumulate
2741 * speed only on the
valid cells of a cell batch.
2744 *
for (
unsigned int v = 0; v < data.n_active_entries_per_cell_batch(cell);
2746 *
for (
unsigned int d = 0;
d < 3; ++
d)
2747 * max_transport =
std::max(max_transport, local_max[v]);
2752 *
return max_transport;
2760 * <a name=
"TheEulerProblemclass"></a>
2761 * <h3>The EulerProblem
class</h3>
2765 * This
class combines the EulerOperator class with the time integrator and
2767 * actually
run the simulations of the Euler problem.
2771 * The member variables are a
triangulation, a finite element, a mapping (to
2772 * create high-order curved surfaces, see
e.g. @ref step_10
"step-10"), and a
DoFHandler to
2773 * describe the degrees of freedom. In addition, we keep an instance of the
2774 * EulerOperator described above around, which will
do all heavy lifting in
2775 * terms of integrals, and some parameters
for time integration like the
2776 * current time or the time step size.
2780 * Furthermore, we use a PostProcessor instance to write some additional
2781 * information to the output file, in similarity to what was done in
2782 * @ref step_33
"step-33". The
interface of the
DataPostprocessor class is intuitive,
2783 * requiring us to provide information about what needs to be evaluated
2784 * (typically only the values of the solution, except
for the Schlieren plot
2785 * that we only enable in 2D where it makes sense), and the names of what
2786 * gets evaluated. Note that it would also be possible to
extract most
2787 * information by calculator tools within visualization programs such as
2788 * ParaView, but it is so much more convenient to
do it already when writing
2792 * template <int dim>
2793 *
class EulerProblem
2801 *
void make_grid_and_dofs();
2803 *
void output_results(
const unsigned int result_number);
2809 * #ifdef DEAL_II_WITH_P4EST
2821 * EulerOperator<dim, fe_degree, n_q_points_1d> euler_operator;
2823 *
double time, time_step;
2832 * std::vector<
Vector<double>> &computed_quantities)
const override;
2834 *
virtual std::vector<std::string>
get_names()
const override;
2836 *
virtual std::vector<
2843 *
const bool do_schlieren_plot;
2849 *
template <
int dim>
2850 * EulerProblem<dim>::Postprocessor::Postprocessor()
2851 * : do_schlieren_plot(dim == 2)
2858 * For the main evaluation of the field variables, we
first check that the
2859 * lengths of the arrays
equal the expected values (the lengths `2*dim+4` or
2860 * `2*dim+5` are derived from the sizes of the names we specify in the
2861 * get_names()
function below). Then we
loop over all evaluation points and
2862 * fill the respective information: First we fill the primal solution
2863 * variables of density @f$\rho@f$, momentum @f$\rho \mathbf{u}@f$ and energy @f$E@f$,
2864 * then we compute the derived velocity @f$\mathbf u@f$, the pressure @f$p@f$, the
2865 * speed of sound @f$c=
\sqrt{
\gamma p / \rho}@f$, as well as the Schlieren plot
2866 * showing @f$s = |\nabla \rho|^2@f$ in
case it is enabled. (See @ref step_69
"step-69" for
2867 * another example where we create a Schlieren plot.)
2870 *
template <
int dim>
2871 *
void EulerProblem<dim>::Postprocessor::evaluate_vector_field(
2875 *
const unsigned int n_evaluation_points = inputs.solution_values.size();
2877 *
if (do_schlieren_plot ==
true)
2878 *
Assert(inputs.solution_gradients.size() == n_evaluation_points,
2881 *
Assert(computed_quantities.size() == n_evaluation_points,
2884 *
Assert(computed_quantities[0].size() ==
2885 * dim + 2 + (do_schlieren_plot ==
true ? 1 : 0),
2888 *
for (
unsigned int q = 0; q < n_evaluation_points; ++q)
2891 *
for (
unsigned int d = 0;
d < dim + 2; ++
d)
2892 * solution[
d] = inputs.solution_values[q](
d);
2894 *
const double density = solution[0];
2896 *
const double pressure = euler_pressure<dim>(solution);
2898 *
for (
unsigned int d = 0;
d < dim; ++
d)
2899 * computed_quantities[q](
d) = velocity[
d];
2900 * computed_quantities[q](dim) = pressure;
2901 * computed_quantities[q](dim + 1) = std::sqrt(
gamma * pressure / density);
2903 *
if (do_schlieren_plot ==
true)
2904 * computed_quantities[q](dim + 2) =
2905 * inputs.solution_gradients[q][0] * inputs.solution_gradients[q][0];
2911 *
template <
int dim>
2912 * std::vector<std::string> EulerProblem<dim>::Postprocessor::get_names() const
2914 * std::vector<std::string> names;
2915 *
for (
unsigned int d = 0;
d < dim; ++
d)
2916 * names.emplace_back(
"velocity");
2917 * names.emplace_back(
"pressure");
2918 * names.emplace_back(
"speed_of_sound");
2920 *
if (do_schlieren_plot ==
true)
2921 * names.emplace_back(
"schlieren_plot");
2930 * For the interpretation of quantities, we have scalar density, energy,
2931 * pressure, speed of sound, and the Schlieren plot, and vectors
for the
2932 * momentum and the velocity.
2935 *
template <
int dim>
2936 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
2937 * EulerProblem<dim>::Postprocessor::get_data_component_interpretation() const
2939 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
2941 *
for (
unsigned int d = 0;
d < dim; ++
d)
2942 * interpretation.push_back(
2947 *
if (do_schlieren_plot ==
true)
2948 * interpretation.push_back(
2951 *
return interpretation;
2958 * With respect to the necessary update flags, we only need the values
for
2959 * all quantities but the Schlieren plot, which is based on the density
2963 *
template <
int dim>
2964 *
UpdateFlags EulerProblem<dim>::Postprocessor::get_needed_update_flags() const
2966 *
if (do_schlieren_plot ==
true)
2976 * The constructor
for this class is unsurprising: We
set up a
parallel
2977 *
triangulation based on the `MPI_COMM_WORLD` communicator, a vector finite
2978 * element with `dim+2` components for density, momentum, and energy, a
2979 * high-order mapping of the same degree as the underlying finite element,
2980 * and initialize the time and time step to
zero.
2983 * template <int dim>
2984 * EulerProblem<dim>::EulerProblem()
2985 * : pcout(std::cout, Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
2986 * #ifdef DEAL_II_WITH_P4EST
2989 * , fe(FE_DGQ<dim>(fe_degree), dim + 2)
2990 * , mapping(fe_degree)
2991 * , dof_handler(triangulation)
2992 * , timer(pcout, TimerOutput::never, TimerOutput::wall_times)
2993 * , euler_operator(timer)
3002 * As a mesh,
this tutorial program implements two options, depending on the
3003 * global variable `testcase`: For the analytical variant (`testcase==0`),
3004 * the domain is @f$(0, 10) \times (-5, 5)@f$, with Dirichlet boundary
3005 * conditions (inflow) all around the domain. For `testcase==1`, we
set the
3006 * domain to a
cylinder in a rectangular box, derived from the flow past
3007 *
cylinder testcase
for incompressible viscous flow by Schäfer and
3008 * Turek (1996). Here, we have a larger variety of boundaries. The inflow
3009 * part at the left of the channel is given the inflow type,
for which we
3010 * choose a constant inflow profile, whereas we
set a subsonic outflow at
3011 * the right. For the boundary around the
cylinder (boundary
id equal to 2)
3012 * as well as the channel walls (boundary
id equal to 3) we use the wall
3013 * boundary type, which is no-normal flow. Furthermore,
for the 3D
cylinder
3014 * we also add a gravity force in vertical direction. Having the base mesh
3015 * in place (including the manifolds
set by
3017 * specified number of global refinements, create the unknown numbering from
3019 * initialization of the EulerOperator.
3022 *
template <
int dim>
3023 *
void EulerProblem<dim>::make_grid_and_dofs()
3030 *
for (
unsigned int d = 1;
d < dim; ++
d)
3031 * lower_left[
d] = -5;
3034 * upper_right[0] = 10;
3035 *
for (
unsigned int d = 1;
d < dim; ++
d)
3036 * upper_right[
d] = 5;
3043 * euler_operator.set_inflow_boundary(
3044 * 0, std_cxx14::make_unique<ExactSolution<dim>>(0));
3054 * euler_operator.set_inflow_boundary(
3055 * 0, std_cxx14::make_unique<ExactSolution<dim>>(0));
3056 * euler_operator.set_subsonic_outflow_boundary(
3057 * 1, std_cxx14::make_unique<ExactSolution<dim>>(0));
3059 * euler_operator.set_wall_boundary(2);
3060 * euler_operator.set_wall_boundary(3);
3063 * euler_operator.set_body_force(
3065 * std::vector<double>({0., 0., -0.2})));
3078 * euler_operator.reinit(mapping, dof_handler);
3079 * euler_operator.initialize_vector(solution);
3083 * In the following, we output some statistics about the problem. Because we
3084 * often
end up with quite large
numbers of cells or degrees of freedom, we
3085 * would like to print them with a comma to separate each
set of three
3086 * digits. This can be done via
"locales", although the way
this works is
3087 * not particularly intuitive. @ref step_32
"step-32" explains
this in slightly more
3091 * std::locale s = pcout.get_stream().getloc();
3092 * pcout.get_stream().imbue(std::locale(
""));
3093 * pcout <<
"Number of degrees of freedom: " << dof_handler.
n_dofs()
3094 * <<
" ( = " << (dim + 2) <<
" [vars] x "
3098 * pcout.get_stream().imbue(s);
3105 * For output, we
first let the Euler
operator compute the errors of the
3106 * numerical results. More precisely, we compute the error against the
3107 * analytical result
for the analytical solution
case, whereas we compute
3108 * the deviation against the background field with constant density and
3109 * energy and constant velocity in @f$x@f$ direction
for the
second test
case.
3113 * The next step is to create output. This is similar to what is done in
3114 * @ref step_33
"step-33": We let the postprocessor defined above control most of the
3115 * output, except
for the primal field that we write directly. For the
3116 * analytical solution test
case, we also perform another projection of the
3117 * analytical solution and print the difference between that field and the
3118 * numerical solution. Once we have defined all quantities to be written, we
3119 * build the patches
for output. Similarly to @ref step_65
"step-65", we create a
3120 * high-order VTK output by setting the appropriate flag, which enables us
3121 * to visualize fields of high polynomial degrees. Finally, we
call the
3123 * to the given file name. This
function uses special MPI
parallel write
3124 * facilities, which are typically more optimized
for parallel file systems
3125 * than the standard library
's `std::ofstream` variants used in most other
3126 * tutorial programs. A particularly nice feature of the
3127 * `write_vtu_in_parallel()` function is the fact that it can combine output
3128 * from all MPI ranks into a single file, obviating a VTU master file (the
3133 * For parallel programs, it is often instructive to look at the partitioning
3134 * of cells among processors. To this end, one can pass a vector of numbers
3135 * to DataOut::add_data_vector() that contains as many entries as the
3136 * current processor has active cells; these numbers should then be the
3137 * rank of the processor that owns each of these cells. Such a vector
3138 * could, for example, be obtained from
3139 * GridTools::get_subdomain_association(). On the other hand, on each MPI
3140 * process, DataOut will only read those entries that correspond to locally
3141 * owned cells, and these of course all have the same value: namely, the rank
3142 * of the current process. What is in the remaining entries of the vector
3143 * doesn't actually matter, and so we can just get away with a cheap trick: We
3145 * with the rank of the current MPI process. The key is that on each process,
3146 * only the entries corresponding to the locally owned cells will be read,
3147 * ignoring the (wrong) values in other entries. The fact that every process
3148 * submits a vector in which the correct subset of entries is correct is all
3149 * that is necessary.
3152 *
template <
int dim>
3153 *
void EulerProblem<dim>::output_results(
const unsigned int result_number)
3155 *
const std::array<double, 3> errors =
3156 * euler_operator.compute_errors(ExactSolution<dim>(time), solution);
3157 *
const std::string quantity_name = testcase == 0 ?
"error" :
"norm";
3159 * pcout <<
"Time:" << std::setw(8) << std::setprecision(3) << time
3160 * <<
", dt: " << std::setw(8) << std::setprecision(2) << time_step
3161 * <<
", " << quantity_name <<
" rho: " << std::setprecision(4)
3162 * << std::setw(10) << errors[0] <<
", rho * u: " << std::setprecision(4)
3163 * << std::setw(10) << errors[1] <<
", energy:" << std::setprecision(4)
3164 * << std::setw(10) << errors[2] << std::endl;
3169 * Postprocessor postprocessor;
3178 * std::vector<std::string> names;
3179 * names.emplace_back(
"density");
3180 *
for (
unsigned int d = 0;
d < dim; ++
d)
3181 * names.emplace_back(
"momentum");
3182 * names.emplace_back(
"energy");
3184 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
3186 * interpretation.push_back(
3188 *
for (
unsigned int d = 0;
d < dim; ++
d)
3189 * interpretation.push_back(
3191 * interpretation.push_back(
3194 * data_out.
add_data_vector(dof_handler, solution, names, interpretation);
3199 *
if (testcase == 0 && dim == 2)
3201 * reference.
reinit(solution);
3202 * euler_operator.project(ExactSolution<dim>(time), reference);
3203 * reference.
sadd(-1., 1, solution);
3204 * std::vector<std::string> names;
3205 * names.emplace_back(
"error_density");
3206 *
for (
unsigned int d = 0;
d < dim; ++
d)
3207 * names.emplace_back(
"error_momentum");
3208 * names.emplace_back(
"error_energy");
3210 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
3212 * interpretation.push_back(
3214 *
for (
unsigned int d = 0;
d < dim; ++
d)
3215 * interpretation.push_back(
3217 * interpretation.push_back(
3234 *
const std::string filename =
3245 * by calling the function that creates the mesh and sets up data structures,
3246 * and then initializing the time integrator and the two temporary vectors of
3247 * the low-storage integrator. We
call these vectors `rk_register_1` and
3248 * `rk_register_2`, and use the
first vector to represent the quantity
3249 * @f$\mathbf{r}_i@f$ and the
second one for @f$\mathbf{k}_i@f$ in the formulas
for
3250 * the Runge--Kutta scheme outlined in the introduction. Before we start the
3251 * time
loop, we compute the time step size by the
3252 * `EulerOperator::compute_cell_transport_speed()`
function. For reasons of
3253 * comparison, we compare the result obtained there with the minimal mesh
3254 * size and print them to screen. For velocities and speeds of sound close
3255 * to unity as in
this tutorial program, the predicted effective mesh size
3256 * will be close, but they could vary
if scaling were different.
3259 *
template <
int dim>
3264 *
const unsigned int n_vect_bits = 8 *
sizeof(Number) * n_vect_number;
3266 * pcout <<
"Running with "
3268 * <<
" MPI processes" << std::endl;
3269 * pcout <<
"Vectorization over " << n_vect_number <<
" "
3271 * <<
" = " << n_vect_bits <<
" bits ("
3276 * make_grid_and_dofs();
3278 *
const LowStorageRungeKuttaIntegrator integrator(lsrk_scheme);
3282 * rk_register_1.
reinit(solution);
3283 * rk_register_2.
reinit(solution);
3285 * euler_operator.project(ExactSolution<dim>(time), solution);
3288 *
for (
const auto &cell :
triangulation.active_cell_iterators())
3289 *
if (cell->is_locally_owned())
3290 * min_vertex_distance =
3291 *
std::min(min_vertex_distance, cell->minimum_vertex_distance());
3292 * min_vertex_distance =
3295 * time_step = courant_number * integrator.n_stages() /
3296 * euler_operator.compute_cell_transport_speed(solution);
3297 * pcout <<
"Time step size: " << time_step
3298 * <<
", minimal h: " << min_vertex_distance
3299 * <<
", initial transport scaling: "
3300 * << 1. / euler_operator.compute_cell_transport_speed(solution)
3304 * output_results(0);
3308 * Now we are ready to start the time
loop, which we
run until the time
3309 * has reached the desired
end time. Every 5 time steps, we compute a
new
3310 * estimate
for the time step -- since the solution is nonlinear, it is
3311 * most effective to adapt the
value during the course of the
3312 * simulation. In
case the Courant number was chosen too aggressively, the
3313 * simulation will typically blow up with time step NaN, so that is easy
3314 * to detect here. One thing to note is that roundoff errors might
3315 * propagate to the leading digits due to an interaction of slightly
3316 * different time step selections that in turn lead to slightly different
3317 * solutions. To decrease
this sensitivity, it is common practice to round
3318 * or truncate the time step size to a few digits,
e.g. 3 in
this case. In
3319 *
case the current time is near the prescribed
'tick' value for output
3320 * (
e.g. 0.02), we also write the output. After the
end of the time
loop,
3321 * we summarize the computation by printing some statistics, which is
3325 *
unsigned int timestep_number = 0;
3327 * while (time < final_time - 1
e-12)
3329 * ++timestep_number;
3330 *
if (timestep_number % 5 == 0)
3332 * courant_number * integrator.n_stages() /
3334 * euler_operator.compute_cell_transport_speed(solution), 3);
3338 * integrator.perform_time_step(euler_operator,
3346 * time += time_step;
3348 *
if (
static_cast<int>(time / output_tick) !=
3349 *
static_cast<int>((time - time_step) / output_tick) ||
3350 * time >= final_time - 1
e-12)
3352 *
static_cast<unsigned int>(std::round(time / output_tick)));
3355 * timer.print_wall_time_statistics(MPI_COMM_WORLD);
3356 * pcout << std::endl;
3365 * The main() function is not surprising and follows what was done in all
3366 * previous MPI programs: As we
run an MPI program, we need to
call `MPI_Init()`
3367 * and `MPI_Finalize()`, which we do through the
3368 *
Utilities::MPI::MPI_InitFinalize data structure. Note that we
run the program
3369 * only with MPI, and
set the thread count to 1.
3372 *
int main(
int argc,
char **argv)
3374 *
using namespace Euler_DG;
3375 *
using namespace dealii;
3383 * EulerProblem<dimension> euler_problem;
3384 * euler_problem.run();
3386 *
catch (std::exception &exc)
3388 * std::cerr << std::endl
3390 * <<
"----------------------------------------------------"
3392 * std::cerr <<
"Exception on processing: " << std::endl
3393 * << exc.what() << std::endl
3394 * <<
"Aborting!" << std::endl
3395 * <<
"----------------------------------------------------"
3402 * std::cerr << std::endl
3404 * <<
"----------------------------------------------------"
3406 * std::cerr <<
"Unknown exception!" << std::endl
3407 * <<
"Aborting!" << std::endl
3408 * <<
"----------------------------------------------------"
3416 <a name=
"Results"></a><h1>Results</h1>
3419 <a name=
"Programoutput"></a><h3>Program output</h3>
3422 Running the program with the
default settings on a machine with 40 processes
3423 produces the following output:
3425 Running with 40 MPI processes
3426 Vectorization over 8 doubles = 512 bits (AVX512)
3427 Number of degrees of freedom: 147,456 ( = 4 [vars] x 1,024 [cells] x 36 [dofs/cell/var] )
3428 Time step size: 0.00689325, minimal h: 0.3125,
initial transport scaling: 0.102759
3430 Time: 0, dt: 0.0069, error rho: 2.76
e-07, rho * u: 1.259
e-06, energy: 2.987
e-06
3431 Time: 1.01, dt: 0.0069, error rho: 1.37
e-06, rho * u: 2.252
e-06, energy: 4.153
e-06
3432 Time: 2.01, dt: 0.0069, error rho: 1.561
e-06, rho * u: 2.43
e-06, energy: 4.493
e-06
3433 Time: 3.01, dt: 0.0069, error rho: 1.714
e-06, rho * u: 2.591
e-06, energy: 4.762
e-06
3434 Time: 4.01, dt: 0.0069, error rho: 1.843
e-06, rho * u: 2.625
e-06, energy: 4.985
e-06
3435 Time: 5.01, dt: 0.0069, error rho: 1.496
e-06, rho * u: 1.961
e-06, energy: 4.142
e-06
3436 Time: 6, dt: 0.0083, error rho: 1.007
e-06, rho * u: 7.119
e-07, energy: 2.972
e-06
3437 Time: 7, dt: 0.0095, error rho: 9.096
e-07, rho * u: 3.786
e-07, energy: 2.626
e-06
3438 Time: 8, dt: 0.0096, error rho: 8.439
e-07, rho * u: 3.338
e-07, energy: 2.43
e-06
3439 Time: 9, dt: 0.0096, error rho: 7.822
e-07, rho * u: 2.984
e-07, energy: 2.248
e-06
3440 Time: 10, dt: 0.0096, error rho: 7.231
e-07, rho * u: 2.666
e-07, energy: 2.074
e-06
3442 +-------------------------------------------+------------------+------------+------------------+
3443 | Total wallclock time elapsed | 2.249s 30 | 2.249s | 2.249s 8 |
3445 | Section | no. calls |
min time rank | avg time |
max time rank |
3446 +-------------------------------------------+------------------+------------+------------------+
3447 | compute errors | 11 | 0.008066s 13 | 0.00952s | 0.01041s 20 |
3448 | compute transport speed | 258 | 0.01012s 13 | 0.05392s | 0.08574s 25 |
3449 | output | 11 | 0.9597s 13 | 0.9613s | 0.9623s 6 |
3450 | rk time stepping total | 1283 | 0.9827s 25 | 1.015s | 1.06s 13 |
3451 | rk_stage - integrals L_h | 6415 | 0.8803s 26 | 0.9198s | 0.9619s 14 |
3452 | rk_stage - inv mass + vec upd | 6415 | 0.05677s 15 | 0.06487s | 0.07597s 13 |
3453 +-------------------------------------------+------------------+------------+------------------+
3456 The program output shows that all errors are small. This is due to the fact
3457 that we use a relatively fine mesh of @f$32^2@f$ cells with polynomials of degree
3458 5
for a solution that is smooth. An interesting pattern shows
for the time
3459 step size: whereas it is 0.0069 up to time 5, it increases to 0.0096
for later
3460 times. The step size increases once the vortex with some motion on top of the
3461 speed of sound (and thus faster propagation) leaves the computational domain
3462 between times 5 and 6.5. After that
point, the flow is simply uniform
3463 in the same direction, and the maximum velocity of the gas is reduced
3464 compared to the previous state where the uniform velocity was overlaid
3465 by the vortex. Our time step formula recognizes
this effect.
3467 The
final block of output shows detailed information about the timing
3468 of individual parts of the programs; it breaks
this down by showing
3469 the time taken by the fastest and the slowest processor, and the
3470 average time --
this is often useful in very large computations to
3471 find whether there are processors that are consistently overheated
3472 (and consequently are throttling their clock speed) or consistently
3473 slow
for other reasons.
3474 The summary shows that 1283 time steps have been performed
3475 in 1.02 seconds (looking at the average time among all MPI processes),
while
3476 the output of 11 files has taken additional 0.96 seconds. Broken down per time
3477 step and into the five Runge--Kutta stages, the compute time per evaluation is
3478 0.16 milliseconds. This high performance is typical of
matrix-
free evaluators
3479 and a reason why
explicit time integration is very competitive against
3480 implicit solvers, especially
for large-
scale simulations. The breakdown of
3481 computational times at the
end of the program
run shows that the evaluation of
3482 integrals in @f$\mathcal L_h@f$ contributes with around 0.92 seconds and the
3483 application of the inverse mass
matrix with 0.06 seconds. Furthermore, the
3484 estimation of the transport speed
for the time step size computation
3485 contributes with another 0.05 seconds of compute time.
3487 If we use three more levels of global refinement and 9.4 million DoFs in total,
3488 the
final statistics are as follows (
for the modified Lax--Friedrichs flux,
3489 @f$p=5@f$, and the same system of 40 cores of dual-socket Intel Xeon Gold 6230):
3491 +-------------------------------------------+------------------+------------+------------------+
3492 | Total wallclock time elapsed | 244.9s 12 | 244.9s | 244.9s 34 |
3494 | Section | no. calls |
min time rank | avg time |
max time rank |
3495 +-------------------------------------------+------------------+------------+------------------+
3496 | compute errors | 11 | 0.4239s 12 | 0.4318s | 0.4408s 9 |
3497 | compute transport speed | 2053 | 3.962s 12 | 6.727s | 10.12s 7 |
3498 | output | 11 | 30.35s 12 | 30.36s | 30.37s 9 |
3499 | rk time stepping total | 10258 | 201.7s 7 | 205.1s | 207.8s 12 |
3500 | rk_stage - integrals L_h | 51290 | 121.3s 6 | 126.6s | 136.3s 16 |
3501 | rk_stage - inv mass + vec upd | 51290 | 66.19s 16 | 77.52s | 81.84s 10 |
3502 +-------------------------------------------+------------------+------------+------------------+
3505 Per time step, the solver now takes 0.02 seconds, about 25 times as long as
3506 for the small problem with 147k unknowns. Given that the problem involves 64
3507 times as many unknowns, the increase in computing time is not
3508 surprising. Since we also do 8 times as many time steps, the compute time
3509 should in theory increase by a factor of 512. The actual increase is 205 s /
3510 1.02 s = 202. This is because the small problem size cannot fully utilize the
3511 40 cores due to communication overhead. This becomes clear if we look into the
3512 details of the operations done per time step. The evaluation of the
3513 differential operator @f$\mathcal L_h@f$ with nearest neighbor communication goes
3514 from 0.92 seconds to 127 seconds, i.
e., it increases with a factor of 138. On
3515 the other hand, the cost for application of the inverse mass
matrix and the
3516 vector updates, which do not need to communicate between the MPI processes at
3517 all, has increased by a factor of 1195. The increase is more than the
3518 theoretical factor of 512 because the operation is limited by the bandwidth
3519 from RAM memory for the larger size while for the smaller size, all vectors
3520 fit into the caches of the CPU. The
numbers show that the mass
matrix
3521 evaluation and vector update part consume almost 40% of the time spent by the
3522 Runge--Kutta stages -- despite using a low-storage Runge--Kutta integrator and
3523 merging of vector operations! And despite using over-integration for the
3524 @f$\mathcal L_h@f$ operator. For simpler differential operators and more expensive
3525 time integrators, the proportion spent in the mass
matrix and vector update
3526 part can also reach 70%. If we compute a throughput number in terms of DoFs
3527 processed per
second and Runge--Kutta stage, we obtain @f[ \text{throughput} =
3528 \frac{n_\mathrm{time steps} n_\mathrm{stages}
3529 n_\mathrm{dofs}}{t_\mathrm{compute}} = \frac{10258 \cdot 5 \cdot
3530 9.4\,\text{MDoFs}}{205s} = 2360\, \text{MDoFs/s} @f] This throughput number is
3531 very high, given that simply copying
one vector to another
one runs at
3532 only around 10,000 MDoFs/s.
3534 If we go to the next-larger size with 37.7 million DoFs, the overall
3535 simulation time is 2196 seconds, with 1978 seconds spent in the time
3536 stepping. The increase in
run time is a factor of 9.3
for the L_h
operator
3537 (1179 versus 127 seconds) and a factor of 10.3
for the inverse mass
matrix and
3538 vector updates (797 vs 77.5 seconds). The reason
for this non-optimal increase
3539 in
run time can be traced back to cache effects on the given hardware (with 40
3540 MB of
L2 cache and 55 MB of L3 cache): While not all of the relevant data fits
3541 into caches for 9.4 million DoFs (
one vector takes 75 MB and we have three
3542 vectors plus some additional data in
MatrixFree), there is capacity for
one and
3543 a half vector nonetheless. Given that modern caches are more sophisticated than
3544 the naive least-recently-used strategy (where we would have little re-use as
3545 the data is used in a streaming-like fashion), we can assume that a sizeable
3546 fraction of data can indeed be delivered from caches for the 9.4 million DoFs
3547 case. For the larger case, even with optimal caching less than 10 percent of
3548 data would fit into caches, with an associated loss in performance.
3551 <a name=
"Convergenceratesfortheanalyticaltestcase"></a><h3>Convergence rates for the analytical test case</h3>
3554 For the modified Lax--Friedrichs flux and measuring the error in the momentum
3555 variable, we obtain the following convergence table (the rates are very
3556 similar for the density and energy variables):
3558 <table align=
"center" class=
"doxtable">
3561 <th colspan=
"3"><i>p</i>=2</th>
3562 <th colspan=
"3"><i>p</i>=3</th>
3563 <th colspan=
"3"><i>p</i>=5</th>
3578 <td align=
"right">16</td>
3585 <td align=
"right">2,304</td>
3586 <td align=
"center">1.373e-01</td>
3590 <td align=
"right">64</td>
3594 <td align=
"right">4,096</td>
3595 <td align=
"center">9.130e-02</td>
3597 <td align=
"right">9,216</td>
3598 <td align=
"center">8.899e-03</td>
3602 <td align=
"right">256</td>
3603 <td align=
"right">9,216</td>
3604 <td align=
"center">5.577e-02</td>
3606 <td align=
"right">16,384</td>
3607 <td align=
"center">7.381e-03</td>
3609 <td align=
"right">36,864</td>
3610 <td align=
"center">2.082e-04</td>
3614 <td align=
"right">1024</td>
3615 <td align=
"right">36,864</td>
3616 <td align=
"center">4.724e-03</td>
3618 <td align=
"right">65,536</td>
3619 <td align=
"center">3.072e-04</td>
3621 <td align=
"right">147,456</td>
3622 <td align=
"center">2.625e-06</td>
3626 <td align=
"right">4096</td>
3627 <td align=
"right">147,456</td>
3628 <td align=
"center">6.205e-04</td>
3630 <td align=
"right">262,144</td>
3631 <td align=
"center">1.880e-05</td>
3633 <td align=
"right">589,824</td>
3634 <td align=
"center">3.268e-08</td>
3638 <td align=
"right">16,384</td>
3639 <td align=
"right">589,824</td>
3640 <td align=
"center">8.279e-05</td>
3642 <td align=
"right">1,048,576</td>
3643 <td align=
"center">1.224e-06</td>
3645 <td align=
"right">2,359,296</td>
3646 <td align=
"center">9.252e-10</td>
3650 <td align=
"right">65,536</td>
3651 <td align=
"right">2,359,296</td>
3652 <td align=
"center">1.105e-05</td>
3654 <td align=
"right">4,194,304</td>
3655 <td align=
"center">7.871e-08</td>
3657 <td align=
"right">9,437,184</td>
3658 <td align=
"center">1.369e-10</td>
3662 <td align=
"right">262,144</td>
3663 <td align=
"right">9,437,184</td>
3664 <td align=
"center">1.615e-06</td>
3666 <td align=
"right">16,777,216</td>
3667 <td align=
"center">4.961e-09</td>
3669 <td align=
"right">37,748,736</td>
3670 <td align=
"center">7.091e-11</td>
3675 If we
switch to the Harten-Lax-van Leer flux, the results are as follows:
3676 <table align=
"center" class=
"doxtable">
3679 <th colspan=
"3"><i>p</i>=2</th>
3680 <th colspan=
"3"><i>p</i>=3</th>
3681 <th colspan=
"3"><i>p</i>=5</th>
3696 <td align=
"right">16</td>
3703 <td align=
"right">2,304</td>
3704 <td align=
"center">1.339e-01</td>
3708 <td align=
"right">64</td>
3712 <td align=
"right">4,096</td>
3713 <td align=
"center">9.037e-02</td>
3715 <td align=
"right">9,216</td>
3716 <td align=
"center">8.849e-03</td>
3720 <td align=
"right">256</td>
3721 <td align=
"right">9,216</td>
3722 <td align=
"center">4.204e-02</td>
3724 <td align=
"right">16,384</td>
3725 <td align=
"center">9.143e-03</td>
3727 <td align=
"right">36,864</td>
3728 <td align=
"center">2.501e-04</td>
3732 <td align=
"right">1024</td>
3733 <td align=
"right">36,864</td>
3734 <td align=
"center">4.913e-03</td>
3736 <td align=
"right">65,536</td>
3737 <td align=
"center">3.257e-04</td>
3739 <td align=
"right">147,456</td>
3740 <td align=
"center">3.260e-06</td>
3744 <td align=
"right">4096</td>
3745 <td align=
"right">147,456</td>
3746 <td align=
"center">7.862e-04</td>
3748 <td align=
"right">262,144</td>
3749 <td align=
"center">1.588e-05</td>
3751 <td align=
"right">589,824</td>
3752 <td align=
"center">2.953e-08</td>
3756 <td align=
"right">16,384</td>
3757 <td align=
"right">589,824</td>
3758 <td align=
"center">1.137e-04</td>
3760 <td align=
"right">1,048,576</td>
3761 <td align=
"center">9.400e-07</td>
3763 <td align=
"right">2,359,296</td>
3764 <td align=
"center">4.286e-10</td>
3768 <td align=
"right">65,536</td>
3769 <td align=
"right">2,359,296</td>
3770 <td align=
"center">1.476e-05</td>
3772 <td align=
"right">4,194,304</td>
3773 <td align=
"center">5.799e-08</td>
3775 <td align=
"right">9,437,184</td>
3776 <td align=
"center">2.789e-11</td>
3780 <td align=
"right">262,144</td>
3781 <td align=
"right">9,437,184</td>
3782 <td align=
"center">2.038e-06</td>
3784 <td align=
"right">16,777,216</td>
3785 <td align=
"center">3.609e-09</td>
3787 <td align=
"right">37,748,736</td>
3788 <td align=
"center">5.730e-11</td>
3793 The tables show that we get optimal @f$\mathcal O\left(h^{p+1}\right)@f$
3794 convergence rates
for both numerical fluxes. The errors are slightly smaller
3795 for the Lax--Friedrichs flux
for @f$p=2@f$, but the picture is reversed
for
3796 @f$p=3@f$; in any
case, the differences on
this testcase are relatively
3799 For @f$p=5@f$, we reach the roundoff accuracy of @f$10^{-11}@f$ with both
3800 fluxes on the finest grids. Also note that the errors are absolute with a
3801 domain length of @f$10^2@f$, so relative errors are below @f$10^{-12}@f$. The HLL flux
3802 is somewhat better
for the highest degree, which is due to a slight inaccuracy
3803 of the Lax--Friedrichs flux: The Lax--Friedrichs flux sets a Dirichlet
3804 condition on the solution that leaves the domain, which results in a small
3805 artificial reflection, which is accentuated
for the Lax--Friedrichs
3806 flux. Apart from that, we see that the influence of the numerical flux is
3807 minor, as the polynomial part inside elements is the main driver of the
3808 accucary. The limited influence of the flux also has consequences when trying
3809 to approach more challenging setups with the higher-order DG setup: Taking
for
3810 example the parameters and grid of @ref step_33
"step-33", we get oscillations (which in turn
3811 make density negative and make the solution explode) with both fluxes once the
3812 high-mass part comes near the boundary, as opposed to the low-order finite
3813 volume case (@f$p=0@f$). Thus, any
case that leads to shocks in the solution
3814 necessitates some form of limiting or artificial dissipation. For another
3815 alternative, see the @ref step_69
"step-69" tutorial program.
3818 <a name=
"Resultsforflowinchannelaroundcylinderin2D"></a><h3>Results
for flow in channel around
cylinder in 2D</h3>
3821 For the test
case of the flow around a
cylinder in a channel, we need to
3822 change the
first code line to
3824 constexpr
unsigned int testcase = 1;
3826 This test
case starts with a background field of a constant velocity
3827 of Mach number 0.31 and a constant
initial density; the flow will have
3828 to go around an obstacle in the form of a
cylinder. Since we impose a
3829 no-penetration condition on the
cylinder walls, the flow that
3830 initially impinges head-on onto to
cylinder has to rearrange,
3831 which creates a big sound wave. The following pictures show the pressure at
3832 times 0.1, 0.25, 0.5, and 1.0 (top left to bottom right)
for the 2D
case with
3833 5 levels of global refinement,
using 102,400 cells with polynomial degree of
3834 5 and 14.7 million degrees of freedom over all 4 solution variables.
3835 We clearly see the discontinuity that
3836 propagates slowly in the upstream direction and more quickly in
downstream
3837 direction in the
first snapshot at time 0.1. At time 0.25, the sound wave has
3838 reached the top and bottom walls and reflected back to the interior. From the
3839 different distances of the reflected waves from lower and upper walls we can
3840 see the slight asymmetry of the Schäfer-Turek test
case represented by
3842 cylinder compared to below. At later times, the picture is more chaotic with
3843 many sound waves all over the place.
3845 <table align=
"center" class=
"doxtable" style=
"width:85%">
3848 <img src=
"https://www.dealii.org/images/steps/developer/step-67.pressure_010.png" alt=
"" width=
"100%">
3851 <img src=
"https://www.dealii.org/images/steps/developer/step-67.pressure_025.png" alt=
"" width=
"100%">
3856 <img src=
"https://www.dealii.org/images/steps/developer/step-67.pressure_050.png" alt=
"" width=
"100%">
3859 <img src=
"https://www.dealii.org/images/steps/developer/step-67.pressure_100.png" alt=
"" width=
"100%">
3864 The next picture shows an elevation plot of the pressure at time 1.0 looking
3865 from the channel inlet towards the outlet at the same resolution -- here,
3866 we can see the large number
3867 of reflections. In the figure, two
types of waves are visible. The
3868 larger-amplitude waves correspond to various reflections that happened as the
3869 initial discontinuity hit the walls, whereas the small-amplitude waves of
3870 size similar to the elements correspond to numerical artifacts. They have their
3871 origin in the finite resolution of the scheme and appear as the discontinuity
3872 travels through elements with high-order polynomials. This effect can be cured
3873 by increasing resolution. Apart from
this effect, the rich wave structure is
3874 the result of the transport accuracy of the high-order DG method.
3876 <img src=
"https://www.dealii.org/images/steps/developer/step-67.pressure_elevated.jpg" alt=
"" width=
"40%">
3878 With 2 levels of global refinement with 1,600 cells, the mesh and its
3879 partitioning on 40 MPI processes looks as follows:
3881 <img src=
"https://www.dealii.org/images/steps/developer/step-67.grid-owner.png" alt=
"" width=
"70%">
3883 When we
run the code with 4 levels of global refinements on 40 cores, we get
3884 the following output:
3886 Running with 40 MPI processes
3887 Vectorization over 8 doubles = 512 bits (AVX512)
3888 Number of degrees of freedom: 3,686,400 ( = 4 [vars] x 25,600 [cells] x 36 [dofs/cell/var] )
3889 Time step size: 7.39876
e-05, minimal h: 0.001875,
initial transport scaling: 0.00110294
3891 Time: 0, dt: 7.4
e-05,
norm rho: 4.17
e-16, rho * u: 1.629
e-16, energy: 1.381
e-15
3892 Time: 0.05, dt: 6.3
e-05,
norm rho: 0.02075, rho * u: 0.03801, energy: 0.08772
3893 Time: 0.1, dt: 5.9
e-05,
norm rho: 0.02211, rho * u: 0.04515, energy: 0.08953
3894 Time: 0.15, dt: 5.7
e-05,
norm rho: 0.02261, rho * u: 0.04592, energy: 0.08967
3895 Time: 0.2, dt: 5.8
e-05,
norm rho: 0.02058, rho * u: 0.04361, energy: 0.08222
3896 Time: 0.25, dt: 5.9
e-05,
norm rho: 0.01695, rho * u: 0.04203, energy: 0.06873
3897 Time: 0.3, dt: 5.9
e-05,
norm rho: 0.01653, rho * u: 0.0401, energy: 0.06604
3898 Time: 0.35, dt: 5.7
e-05,
norm rho: 0.01774, rho * u: 0.04264, energy: 0.0706
3902 Time: 1.95, dt: 5.8
e-05,
norm rho: 0.01488, rho * u: 0.03923, energy: 0.05185
3903 Time: 2, dt: 5.7
e-05,
norm rho: 0.01432, rho * u: 0.03969, energy: 0.04889
3905 +-------------------------------------------+------------------+------------+------------------+
3906 | Total wallclock time elapsed | 273.6s 13 | 273.6s | 273.6s 0 |
3908 | Section | no. calls |
min time rank | avg time |
max time rank |
3909 +-------------------------------------------+------------------+------------+------------------+
3910 | compute errors | 41 | 0.01112s 35 | 0.0672s | 0.1337s 0 |
3911 | compute transport speed | 6914 | 5.422s 35 | 15.96s | 29.99s 1 |
3912 | output | 41 | 37.24s 35 | 37.3s | 37.37s 0 |
3913 | rk time stepping total | 34564 | 205.4s 1 | 219.5s | 230.1s 35 |
3914 | rk_stage - integrals L_h | 172820 | 153.6s 1 | 164.9s | 175.6s 27 |
3915 | rk_stage - inv mass + vec upd | 172820 | 47.13s 13 | 53.09s | 64.05s 33 |
3916 +-------------------------------------------+------------------+------------+------------------+
3919 The norms shown here
for the various quantities are the deviations
3920 @f$\rho
'@f$, @f$(\rho u)'@f$, and @f$E
'@f$ against the background field (namely, the
3921 initial condition). The distribution of run time is overall similar as in the
3922 previous test case. The only slight difference is the larger proportion of
3923 time spent in @f$\mathcal L_h@f$ as compared to the inverse mass matrix and vector
3924 updates. This is because the geometry is deformed and the matrix-free
3925 framework needs to load additional arrays for the geometry from memory that
3926 are compressed in the affine mesh case.
3928 Increasing the number of global refinements to 5, the output becomes:
3930 Running with 40 MPI processes
3931 Vectorization over 8 doubles = 512 bits (AVX512)
3932 Number of degrees of freedom: 14,745,600 ( = 4 [vars] x 102,400 [cells] x 36 [dofs/cell/var] )
3936 +-------------------------------------------+------------------+------------+------------------+
3937 | Total wallclock time elapsed | 2693s 32 | 2693s | 2693s 23 |
3939 | Section | no. calls | min time rank | avg time | max time rank |
3940 +-------------------------------------------+------------------+------------+------------------+
3941 | compute errors | 41 | 0.04537s 32 | 0.173s | 0.3489s 0 |
3942 | compute transport speed | 13858 | 40.75s 32 | 85.99s | 149.8s 0 |
3943 | output | 41 | 153.8s 32 | 153.9s | 154.1s 0 |
3944 | rk time stepping total | 69284 | 2386s 0 | 2450s | 2496s 32 |
3945 | rk_stage - integrals L_h | 346420 | 1365s 32 | 1574s | 1718s 19 |
3946 | rk_stage - inv mass + vec upd | 346420 | 722.5s 10 | 870.7s | 1125s 32 |
3947 +-------------------------------------------+------------------+------------+------------------+
3950 The effect on performance is similar to the analytical test case -- in
3951 theory, computation times should increase by a factor of 8, but we actually
3952 see an increase by a factor of 11 for the time steps (219.5 seconds versus
3953 2450 seconds). This can be traced back to caches, with the small case mostly
3954 fitting in caches. An interesting effect, typical of programs with a mix of
3955 local communication (integrals @f$\mathcal L_h@f$) and global communication (computation of
3956 transport speed) with some load imbalance, can be observed by looking at the
3957 MPI ranks that encounter the minimal and maximal time of different phases,
3958 respectively. Rank 0 reports the fastest throughput for the "rk time stepping
3959 total" part. At the same time, it appears to be slowest for the "compute
3960 transport speed" part, almost a factor of 2 slower than the
3961 average and almost a factor of 4 compared to the faster rank.
3962 Since the latter involves global communication, we can attribute the
3963 slowness in this part to the fact that the local Runge--Kutta stages have
3964 advanced more quickly on this rank and need to wait until the other processors
3965 catch up. At this point, one can wonder about the reason for this imbalance:
3966 The number of cells is almost the same on all MPI processes.
3967 However, the matrix-free framework is faster on affine and Cartesian
3968 cells located towards the outlet of the channel, to which the lower MPI ranks
3969 are assigned. On the other hand, rank 32, which reports the highest run time
3970 for the Runga--Kutta stages, owns the curved cells near the cylinder, for
3971 which no data compression is possible. To improve throughput, we could assign
3972 different weights to different cell types when partitioning the
3973 parallel::distributed::Triangulation object, or even measure the run time for a
3974 few time steps and try to rebalance then.
3976 The throughput per Runge--Kutta stage can be computed to 2085 MDoFs/s for the
3977 14.7 million DoFs test case over the 346,000 Runge--Kutta stages, slightly slower
3978 than the Cartesian mesh throughput of 2360 MDoFs/s reported above.
3980 Finally, if we add one additional refinement, we record the following output:
3982 Running with 40 MPI processes
3983 Vectorization over 8 doubles = 512 bits (AVX512)
3984 Number of degrees of freedom: 58,982,400 ( = 4 [vars] x 409,600 [cells] x 36 [dofs/cell/var] )
3988 Time: 1.95, dt: 1.4e-05, norm rho: 0.01488, rho * u: 0.03923, energy: 0.05183
3989 Time: 2, dt: 1.4e-05, norm rho: 0.01431, rho * u: 0.03969, energy: 0.04887
3991 +-------------------------------------------+------------------+------------+------------------+
3992 | Total wallclock time elapsed | 2.166e+04s 26 | 2.166e+04s | 2.166e+04s 24 |
3994 | Section | no. calls | min time rank | avg time | max time rank |
3995 +-------------------------------------------+------------------+------------+------------------+
3996 | compute errors | 41 | 0.1758s 30 | 0.672s | 1.376s 1 |
3997 | compute transport speed | 27748 | 321.3s 34 | 678.8s | 1202s 1 |
3998 | output | 41 | 616.3s 32 | 616.4s | 616.4s 34 |
3999 | rk time stepping total | 138733 | 1.983e+04s 1 | 2.036e+04s | 2.072e+04s 34 |
4000 | rk_stage - integrals L_h | 693665 | 1.052e+04s 32 | 1.248e+04s | 1.387e+04s 19 |
4001 | rk_stage - inv mass + vec upd | 693665 | 6404s 10 | 7868s | 1.018e+04s 32 |
4002 +-------------------------------------------+------------------+------------+------------------+
4005 The "rk time stepping total" part corresponds to a throughput of 2010 MDoFs/s. The
4006 overall run time to perform 139k time steps is 20k seconds (5.7 hours) or 7
4007 time steps per second -- not so bad for having nearly 60 million
4008 unknowns. More throughput can be achieved by adding more cores to
4012 <a name="Resultsforflowinchannelaroundcylinderin3D"></a><h3>Results for flow in channel around cylinder in 3D</h3>
4015 Switching the channel test case to 3D with 3 global refinements, the output is
4017 Running with 40 MPI processes
4018 Vectorization over 8 doubles = 512 bits (AVX512)
4019 Number of degrees of freedom: 221,184,000 ( = 5 [vars] x 204,800 [cells] x 216 [dofs/cell/var] )
4023 Time: 1.95, dt: 0.00011, norm rho: 0.01131, rho * u: 0.03056, energy: 0.04091
4024 Time: 2, dt: 0.00011, norm rho: 0.0119, rho * u: 0.03142, energy: 0.04425
4026 +-------------------------------------------+------------------+------------+------------------+
4027 | Total wallclock time elapsed | 1.734e+04s 4 | 1.734e+04s | 1.734e+04s 38 |
4029 | Section | no. calls | min time rank | avg time | max time rank |
4030 +-------------------------------------------+------------------+------------+------------------+
4031 | compute errors | 41 | 0.6551s 34 | 3.216s | 7.281s 0 |
4032 | compute transport speed | 3546 | 160s 34 | 393.2s | 776.9s 0 |
4033 | output | 41 | 1350s 34 | 1353s | 1357s 0 |
4034 | rk time stepping total | 17723 | 1.519e+04s 0 | 1.558e+04s | 1.582e+04s 34 |
4035 | rk_stage - integrals L_h | 88615 | 1.005e+04s 32 | 1.126e+04s | 1.23e+04s 11 |
4036 | rk_stage - inv mass + vec upd | 88615 | 3056s 11 | 4322s | 5759s 32 |
4037 +-------------------------------------------+------------------+------------+------------------+
4040 The physics are similar to the 2D case, with a slight motion in the z
4041 direction due to the gravitational force. The throughput per Runge--Kutta
4042 stage in this case is
4044 \text{throughput} = \frac{n_\mathrm{time steps} n_\mathrm{stages}
4045 n_\mathrm{dofs}}{t_\mathrm{compute}} =
4046 \frac{17723 \cdot 5 \cdot 221.2\,\text{M}}{15580s} = 1258\, \text{MDoFs/s}.
4049 The throughput is lower than in 2D because the computation of the @f$\mathcal L_h@f$ term
4050 is more expensive. This is due to over-integration with `degree+2` points and
4051 the larger fraction of face integrals (worse volume-to-surface ratio) with
4052 more expensive flux computations. If we only consider the inverse mass matrix
4053 and vector update part, we record a throughput of 4857 MDoFs/s for the 2D case
4054 of the isentropic vortex with 37.7 million unknowns, whereas the 3D case
4055 runs with 4535 MDoFs/s. The performance is similar because both cases are in
4056 fact limited by the memory bandwidth.
4058 If we go to four levels of global refinement, we need to increase the number
4059 of processes to fit everything in memory -- the computation needs around 350
4060 GB of RAM memory in this case. Also, the time it takes to complete 35k time
4061 steps becomes more tolerable by adding additional resources. We therefore use
4062 6 nodes with 40 cores each, resulting in a computation with 240 MPI processes:
4064 Running with 240 MPI processes
4065 Vectorization over 8 doubles = 512 bits (AVX512)
4066 Number of degrees of freedom: 1,769,472,000 ( = 5 [vars] x 1,638,400 [cells] x 216 [dofs/cell/var] )
4070 Time: 1.95, dt: 5.6e-05, norm rho: 0.01129, rho * u: 0.0306, energy: 0.04086
4071 Time: 2, dt: 5.6e-05, norm rho: 0.01189, rho * u: 0.03145, energy: 0.04417
4073 +-------------------------------------------+------------------+------------+------------------+
4074 | Total wallclock time elapsed | 5.396e+04s 151 | 5.396e+04s | 5.396e+04s 0 |
4076 | Section | no. calls | min time rank | avg time | max time rank |
4077 +-------------------------------------------+------------------+------------+------------------+
4078 | compute errors | 41 | 2.632s 178 | 7.221s | 16.56s 0 |
4079 | compute transport speed | 7072 | 714s 193 | 1553s | 3351s 0 |
4080 | output | 41 | 8065s 176 | 8070s | 8079s 0 |
4081 | rk time stepping total | 35350 | 4.25e+04s 0 | 4.43e+04s | 4.515e+04s 193 |
4082 | rk_stage - integrals L_h | 176750 | 2.936e+04s 134 | 3.222e+04s | 3.67e+04s 99 |
4083 | rk_stage - inv mass + vec upd | 176750 | 7004s 99 | 1.207e+04s | 1.55e+04s 132 |
4084 +-------------------------------------------+------------------+------------+------------------+
4086 This simulation had nearly 2 billion unknowns -- quite a large
4087 computation indeed, and still only needed around 1.5 seconds per time
4091 <a name="Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
4094 The code presented here straight-forwardly extends to adaptive meshes, given
4095 appropriate indicators for setting the refinement flags. Large-scale
4096 adaptivity of a similar solver in the context of the acoustic wave equation
4097 has been achieved by the <a href="https://github.com/kronbichler/exwave">exwave
4098 project</a>. However, in the present context, the benefits of adaptivity are often
4099 limited to early times and effects close to the origin of sound waves, as the
4100 waves eventually reflect and diffract. This leads to steep gradients all over
4101 the place, similar to turbulent flow, and a more or less globally
4104 Another topic that we did not discuss in the results section is a comparison
4105 of different time integration schemes. The program provides four variants of
4106 low-storage Runga--Kutta integrators that each have slightly different
4107 accuracy and stability behavior. Among the schemes implemented here, the
4108 higher-order ones provide additional accuracy but come with slightly lower
4109 efficiency in terms of step size per stage before they violate the CFL
4110 condition. An interesting extension would be to compare the low-storage
4111 variants proposed here with standard Runge--Kutta integrators or to use vector
4112 operations that are run separate from the mass matrix operation and compare
4116 <a name="Moreadvancednumericalfluxfunctionsandskewsymmetricformulations"></a><h4>More advanced numerical flux functions and skew-symmetric formulations</h4>
4119 As mentioned in the introduction, the modified Lax--Friedrichs flux and the
4120 HLL flux employed in this program are only two variants of a large body of
4121 numerical fluxes available in the literature on the Euler equations. One
4122 example is the HLLC flux (Harten-Lax-van Leer-Contact) flux which adds the
4123 effect of rarefaction waves missing in the HLL flux, or the Roe flux. As
4124 mentioned in the introduction, the effect of numerical fluxes on high-order DG
4125 schemes is debatable (unlike for the case of low-order discretizations).
4127 A related improvement to increase the stability of the solver is to also
4128 consider the spatial integral terms. A shortcoming in the rather naive
4129 implementation used above is the fact that the energy conservation of the
4130 original Euler equations (in the absence of shocks) only holds up to a
4131 discretization error. If the solution is under-resolved, the discretization
4132 error can give rise to an increase in the numerical energy and eventually
4133 render the discretization unstable. This is because of the inexact numerical
4134 integration of the terms in the Euler equations, which both contain rational
4135 nonlinearities and higher-degree content from curved cells. A way out of this
4136 dilemma are so-called skew-symmetric formulations, see @cite Gassner2013 for a
4137 simple variant. Skew symmetry means that switching the role of the solution
4138 @f$\mathbf{w}@f$ and test functions @f$\mathbf{v}@f$ in the weak form produces the
4139 exact negative of the original quantity, apart from some boundary terms. In
4140 the discrete setting, the challenge is to keep this skew symmetry also when
4141 the integrals are only computed approximately (in the continuous case,
4142 skew-symmetry is a consequence of integration by parts). Skew-symmetric
4143 numerical schemes balance spatial derivatives in the conservative form
4144 @f$(\nabla \mathbf v, \mathbf{F}(\mathbf w))_{K}@f$ with contributions in the
4145 convective form @f$(\mathbf v, \tilde{\mathbf{F}}(\mathbf w)\nabla
4146 \mathbf{w})_{K}@f$ for some @f$\tilde{\mathbf{F}}@f$. The precise terms depend on
4147 the equation and the integration formula, and can in some cases by understood
4148 by special skew-symmetric finite difference schemes.
4151 <a name="Equippingthecodeforsupersoniccalculations"></a><h4>Equipping the code for supersonic calculations</h4>
4154 As mentioned in the introduction, the solution to the Euler equations develops
4155 shocks as the Mach number increases, which require additional mechanisms to
4156 stabilize the scheme, e.g. in the form of limiters. The main challenge besides
4157 actually implementing the limiter or artificial viscosity approach would be to
4158 load-balance the computations, as the additional computations involved for
4159 limiting the oscillations in troubled cells would make them more expensive than the
4160 plain DG cells without limiting. Furthermore, additional numerical fluxes that
4161 better cope with the discontinuities would also be an option.
4163 One ingredient also necessary for supersonic flows are appropriate boundary
4164 conditions. As opposed to the subsonic outflow boundaries discussed in the
4165 introduction and implemented in the program, all characteristics are outgoing
4166 for supersonic outflow boundaries, so we do not want to prescribe any external
4169 \mathbf{w}^+ = \mathbf{w}^- = \begin{pmatrix} \rho^-\\
4170 (\rho \mathbf u)^- \\ E^-\end{pmatrix} \quad
4174 In the code, we would simply add the additional statement
4176 else if (supersonic_outflow_boundaries.find(boundary_id) !=
4177 supersonic_outflow_boundaries.end())
4183 in the `local_apply_boundary_face()` function.
4185 <a name="ExtensiontothelinearizedEulerequations"></a><h4>Extension to the linearized Euler equations</h4>
4188 When the interest with an Euler solution is mostly in the propagation of sound
4189 waves, it often makes sense to linearize the Euler equations around a
4190 background state, i.e., a given density, velocity and energy (or pressure)
4191 field, and only compute the change against these fields. This is the setting
4192 of the wide field of aeroacoustics. Even though the resolution requirements
4193 are sometimes considerably reduced, implementation gets somewhat more
4194 complicated as the linearization gives rise to additional terms. From a code
4195 perspective, in the operator evaluation we also need to equip the code with
4196 the state to linearize against. This information can be provided either by
4197 analytical functions (that are evaluated in terms of the position of the
4198 quadrature points) or by a vector similar to the solution. Based on that
4199 vector, we would create an additional FEEvaluation object to read from it and
4200 provide the values of the field at quadrature points. If the background
4201 velocity is zero and the density is constant, the linearized Euler equations
4202 further simplify and can equivalently be written in the form of the
4203 acoustic wave equation.
4205 A challenge in the context of sound propagation is often the definition of
4206 boundary conditions, as the computational domain needs to be of finite size,
4207 whereas the actual simulation often spans an infinite (or at least much
4208 larger) physical domain. Conventional Dirichlet or Neumann boundary conditions
4209 give rise to reflections of the sound waves that eventually propagate back to
4210 the region of interest and spoil the solution. Therefore, various variants of
4211 non-reflecting boundary conditions or sponge layers, often in the form of
4213 href="https://en.wikipedia.org/wiki/Perfectly_matched_layer">perfectly
4214 matched layers</a> -- where the solution is damped without reflection
4218 <a name="ExtensiontothecompressibleNavierStokesequation"></a><h4>Extension to the compressible Navier-Stokes equation</h4>
4221 The solver presented in this tutorial program can also be extended to the
4222 compressible Navier--Stokes equations by adding viscous terms, as described in
4223 @cite FehnWallKronbichler2019. To keep as much of the performance obtained
4224 here despite the additional cost of elliptic terms, e.g. via an interior
4225 penalty method, one can switch the basis from FE_DGQ to FE_DGQHermite like in
4226 the @ref step_59 "step-59" tutorial program.
4229 <a name="PlainProg"></a>
4230 <h1> The plain program</h1>
4231 @include "step-67.cc"