 |
Reference documentation for deal.II version 9.5.0
|
Loading...
Searching...
No Matches
|
Reference documentation for deal.II version 9.5.0
|
#include <deal.II/sundials/ida.h>
Classes | |
| class | AdditionalData |
Public Member Functions | |
| IDA (const AdditionalData &data=AdditionalData()) | |
| IDA (const AdditionalData &data, const MPI_Comm mpi_comm) | |
| ~IDA () | |
| unsigned int | solve_dae (VectorType &solution, VectorType &solution_dot) |
| void | reset (const double t, const double h, VectorType &y, VectorType &yp) |
Static Public Member Functions | |
| static ::ExceptionBase & | ExcIDAError (int arg1) |
Public Attributes | |
| std::function< void(VectorType &)> | reinit_vector |
| std::function< void(const double t, const VectorType &y, const VectorType &y_dot, VectorType &res)> | residual |
| std::function< void(const double t, const VectorType &y, const VectorType &y_dot, const double alpha)> | setup_jacobian |
| std::function< void(const VectorType &rhs, VectorType &dst)> | solve_jacobian_system |
| std::function< void(const VectorType &rhs, VectorType &dst, const double tolerance)> | solve_with_jacobian |
| std::function< void(const double t, const VectorType &sol, const VectorType &sol_dot, const unsigned int step_number)> | output_step |
| std::function< bool(const double t, VectorType &sol, VectorType &sol_dot)> | solver_should_restart |
| std::function< IndexSet()> | differential_components |
| std::function< VectorType &()> | get_local_tolerances |
Private Member Functions | |
| void | set_functions_to_trigger_an_assert () |
Static Private Member Functions | |
| static ::ExceptionBase & | ExcFunctionNotProvided (std::string arg1) |
Private Attributes | |
| const AdditionalData | data |
| void * | ida_mem |
| SUNContext | ida_ctx |
| MPI_Comm | mpi_communicator |
| GrowingVectorMemory< VectorType > | mem |
| std::exception_ptr | pending_exception |
Interface to SUNDIALS Implicit Differential-Algebraic (IDA) solver.
The class IDA is a wrapper to SUNDIALS Implicit Differential-Algebraic solver which is a general purpose solver for systems of Differential-Algebraic Equations (DAEs).
The user has to provide the implementation of the following std::functions:
The function solve_jacobian_system() is deprecated. You should use solve_with_jacobian() to leverage better non-linear algorithms.
Optionally, also the following functions could be provided. By default they do nothing, or are not required. If you call the constructor in a way that requires a not-implemented function, an Assertion will be thrown.
To output steps, connect a function to the signal
Citing from the SUNDIALS documentation:
Consider a system of Differential-Algebraic Equations written in the general form
\[ \begin{cases} F(t,y,\dot y) = 0\, , \\ y(t_0) = y_0\, , \\ \dot y (t_0) = \dot y_0\, . \end{cases} \]
where \(y,\dot y\) are vectors in \(\mathbb{R}^n\), \(t\) is often the time (but can also be a parametric quantity), and \(F:\mathbb{R}\times\mathbb{R}^n\times \mathbb{R}^n\rightarrow\mathbb{R}^n\). Such problem is solved using Newton iteration augmented with a line search global strategy. The integration method used in IDA is the variable-order, variable-coefficient BDF (Backward Differentiation Formula), in fixed-leading-coefficient. The method order ranges from 1 to 5, with the BDF of order \(q\) given by the multistep formula
\[ \sum_{i=0}^q \alpha_{n,i}\,y_{n-i}=h_n\,\dot y_n\, , \label{eq:bdf} \]
where \(y_n\) and \(\dot y_n\) are the computed approximations of \(y(t_n)\) and \(\dot y(t_n)\), respectively, and the step size is \(h_n=t_n-t_{n-1}\). The coefficients \(\alpha_{n,i}\) are uniquely determined by the order \(q\), and the history of the step sizes. The application of the BDF method to the DAE system results in a nonlinear algebraic system to be solved at each time step:
\[ G(y_n)\equiv F\left(t_n,y_n,\dfrac{1}{h_n}\sum_{i=0}^q \alpha_{n,i}\,y_{n-i}\right)=0\, . \]
The Newton method leads to a linear system of the form
\[ J[y_{n(m+1)}-y_{n(m)}]=-G(y_{n(m)})\, , \]
where \(y_{n(m)}\) is the \(m\)-th approximation to \(y_n\), and \(J\) is the approximation of the system Jacobian
\[ J=\dfrac{\partial G}{\partial y} = \dfrac{\partial F}{\partial y} + \alpha \dfrac{\partial F}{\partial \dot y}\, , \]
and \(\alpha = \alpha_{n,0}/h_n\). It is worth mentioning that the scalar \(\alpha\) changes whenever the step size or method order changes.
To provide a simple example, consider the following harmonic oscillator problem:
\[ \begin{split} u'' & = -k^2 u \\ u (0) & = 0 \\ u'(0) & = k \end{split} \]
We write it in terms of a first order ode:
\[ \begin{matrix} y_0' & -y_1 & = 0 \\ y_1' & + k^2 y_0 & = 0 \end{matrix} \]
That is \(F(y', y, t) = y' + A y = 0 \) where A =
\[ \begin{pmatrix} 0 & -1 \\ k^2 &0 \end{pmatrix} \]
and \(y(0)=(0, k)\), \(y'(0) = (k, 0)\).
The exact solution is \(y_0(t) = \sin(k t)\), \(y_1(t) = y_0'(t) = k \cos(k t)\), \(y_1'(t) = -k^2 \sin(k t)\).
The Jacobian to assemble is the following: \(J = \alpha I + A\).
This is achieved by the following snippet of code:
| SUNDIALS::IDA< VectorType >::IDA | ( | const AdditionalData & | data = AdditionalData() | ) |
Constructor. It is possible to fine tune the SUNDIALS IDA solver by passing an AdditionalData() object that sets all of the solver parameters.
IDA is a Differential Algebraic solver. As such, it requires initial conditions also for the first order derivatives. If you do not provide consistent initial conditions, (i.e., conditions for which F(y_dot(0), y(0), 0) = 0), you can ask SUNDIALS to compute initial conditions for you by using the ic_type parameter at construction time.
You have three options
By default, this class assumes that all components are differential, and that you want to solve a standard ode. In this case, the initial component type is set to use_y_diff, so that the y_dot at time t=initial_time is computed by solving the nonlinear problem \(F(y_dot,
y(t0), t0) = 0\) in the variable y_dot.
Notice that a Newton solver is used for this computation. The Newton solver parameters can be tweaked by acting on ic_alpha and ic_max_iter.
If you reset the solver at some point, you may want to select a different computation for the initial conditions after reset. Say, for example, that you have refined a grid, and after transferring the solution to the new grid, the initial conditions are no longer consistent. Then you can choose how these are made consistent, using the same three options that you used for the initial conditions in reset_type.
| data | IDA configuration data |
| SUNDIALS::IDA< VectorType >::IDA | ( | const AdditionalData & | data, |
| const MPI_Comm | mpi_comm | ||
| ) |
| SUNDIALS::IDA< VectorType >::~IDA |
| unsigned int SUNDIALS::IDA< VectorType >::solve_dae | ( | VectorType & | solution, |
| VectorType & | solution_dot | ||
| ) |
| void SUNDIALS::IDA< VectorType >::reset | ( | const double | t, |
| const double | h, | ||
| VectorType & | y, | ||
| VectorType & | yp | ||
| ) |
Clear internal memory and start with clean objects. This function is called when the simulation start and when the user returns true to a call to solver_should_restart().
By default solver_should_restart() returns false. If the user needs to implement, for example, local adaptivity in space, he or she may assign a different function to solver_should_restart() that performs all mesh changes, transfers the solution and the solution dot to the new mesh, and returns true.
During reset(), both y and yp are checked for consistency, and according to what was specified as ic_type (if t==initial_time) or reset_type (if t>initial_time), yp, y, or both are modified to obtain a consistent set of initial data.
| [in] | t | The new starting time |
| [in] | h | The new (tentative) starting time step |
| [in,out] | y | The new (tentative) initial solution |
| [in,out] | yp | The new (tentative) initial solution_dot |
|
private |
| std::function<void(VectorType &)> SUNDIALS::IDA< VectorType >::reinit_vector |
| std::function<void(const double t, const VectorType &y, const VectorType &y_dot, VectorType & res)> SUNDIALS::IDA< VectorType >::residual |
Compute residual. Return \(F(t, y, \dot y)\).
| std::function<void(const double t, const VectorType &y, const VectorType &y_dot, const double alpha)> SUNDIALS::IDA< VectorType >::setup_jacobian |
Compute Jacobian. This function is called by IDA any time a Jacobian update is required. The user should compute the Jacobian (or update all the variables that allow the application of the Jacobian). This function is called by IDA once, before any call to solve_jacobian_system() or solve_with_jacobian().
The Jacobian \(J\) should be a (possibly inexact) computation of
\[ J=\dfrac{\partial G}{\partial y} = \dfrac{\partial F}{\partial y} + \alpha \dfrac{\partial F}{\partial \dot y}. \]
If the user uses a matrix based computation of the Jacobian, then this is the right place where an assembly routine should be called to assemble both a matrix and a preconditioner for the Jacobian system. Subsequent calls (possibly more than one) to solve_jacobian_system() or solve_with_jacobian() can assume that this function has been called at least once.
Notice that no assumption is made by this interface on what the user should do in this function. IDA only assumes that after a call to setup_jacobian() it is possible to call solve_jacobian_system() or solve_with_jacobian() to obtain a solution \(x\) to the system \(J x = b\).
| std::function<void(const VectorType &rhs, VectorType &dst)> SUNDIALS::IDA< VectorType >::solve_jacobian_system |
Solve the Jacobian linear system. This function will be called by IDA (possibly several times) after setup_jacobian() has been called at least once. IDA tries to do its best to call setup_jacobian() the minimum amount of times. If convergence can be achieved without updating the Jacobian, then IDA does not call setup_jacobian() again. If, on the contrary, internal IDA convergence tests fail, then IDA calls again setup_jacobian() with updated vectors and coefficients so that successive calls to solve_jacobian_systems() lead to better convergence in the Newton process.
The jacobian \(J\) should be (an approximation of) the system Jacobian
\[ J=\dfrac{\partial G}{\partial y} = \dfrac{\partial F}{\partial y} + \alpha \dfrac{\partial F}{\partial \dot y}. \]
A call to this function should store in dst the result of \(J^{-1}\) applied to src, i.e., J*dst = src. It is the users responsibility to set up proper solvers and preconditioners inside this function.
| std::function< void(const VectorType &rhs, VectorType &dst, const double tolerance)> SUNDIALS::IDA< VectorType >::solve_with_jacobian |
Solve the Jacobian linear system up to a specified tolerance. This function will be called by IDA (possibly several times) after setup_jacobian() has been called at least once. IDA tries to do its best to call setup_jacobian() the minimum number of times. If convergence can be achieved without updating the Jacobian, then IDA does not call setup_jacobian() again. If, on the contrary, internal IDA convergence tests fail, then IDA calls again setup_jacobian() with updated vectors and coefficients so that successive calls to solve_with_jacobian() lead to better convergence in the Newton process.
The Jacobian \(J\) should be (an approximation of) the system Jacobian
\[ J=\dfrac{\partial G}{\partial y} = \dfrac{\partial F}{\partial y} + \alpha \dfrac{\partial F}{\partial \dot y}. \]
Arguments to the function are:
| [in] | rhs | The system right hand side to solve for. |
| [out] | dst | The solution of \(J^{-1} * src\). |
| [in] | tolerance | The tolerance with which to solve the linear system of equations. |
A call to this function should store in dst the result of \(J^{-1}\) applied to src, i.e., the solution of the linear system J*dst = src. It is the user's responsibility to set up proper solvers and preconditioners either inside this function, or already within the setup_jacobian() function. (The latter is, for example, what the step-77 program does: All expensive operations happen in setup_jacobian(), given that that function is called far less often than the current one.)
| std::function<void(const double t, const VectorType & sol, const VectorType & sol_dot, const unsigned int step_number)> SUNDIALS::IDA< VectorType >::output_step |
Process solution. This function is called by IDA at fixed time steps, every output_period seconds, and it is passed a polynomial interpolation of the solution and of its time derivative, computed using the current BDF order and the (internally stored) previously computed solution steps.
Notice that it is well possible that internally IDA computes a time step which is much larger than the output_period step, and therefore calls this function consecutively several times by simply performing all intermediate interpolations. There is no relationship between how many times this function is called and how many time steps have actually been computed.
| std::function<bool(const double t, VectorType &sol, VectorType &sol_dot)> SUNDIALS::IDA< VectorType >::solver_should_restart |
Evaluate whether the solver should be restarted (for example because the number of degrees of freedom has changed).
This function is supposed to perform all operations that are necessary in sol and sol_dot to make sure that the resulting vectors are consistent, and of the correct final size.
For example, one may decide that a local refinement is necessary at time t. This function should then return true, and change the dimension of both sol and sol_dot to reflect the new dimension. Since IDA does not know about the new dimension, an internal reset is necessary.
The default implementation simply returns false, i.e., no restart is performed during the evolution.
| std::function<IndexSet()> SUNDIALS::IDA< VectorType >::differential_components |
Return an index set containing the differential components. Implementation of this function is optional. The default is to return a complete index set. If your equation is also algebraic (i.e., it contains algebraic constraints, or Lagrange multipliers), you should overwrite this function in order to return only the differential components of your system.
When running in parallel, every process will call this function independently, and synchronization will happen at the end of the initialization setup to communicate what components are local. Make sure you only return the locally owned (or locally relevant) components, in order to minimize communication between processes.
| std::function<VectorType &()> SUNDIALS::IDA< VectorType >::get_local_tolerances |
|
private |
|
private |
|
private |
|
private |
|
private |
|
mutableprivate |
1.9.7