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Reference documentation for deal.II version 9.0.0
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Reference documentation for deal.II version 9.0.0
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#include <deal.II/algorithms/theta_timestepping.h>
Public Member Functions | |
| ThetaTimestepping (OperatorBase &op_explicit, OperatorBase &op_implicit) | |
| virtual void | operator() (AnyData &out, const AnyData &in) |
| virtual void | notify (const Event &) |
| void | set_output (OutputOperator< VectorType > &output) |
| void | parse_parameters (ParameterHandler ¶m) |
| double | current_time () const |
| double | theta () const |
| double | theta (double new_theta) |
| const TimestepData & | explicit_data () const |
| const TimestepData & | implicit_data () const |
| TimestepControl & | timestep_control () |
 Public Member Functions inherited from Algorithms::OperatorBase | |
| virtual | ~OperatorBase ()=default |
| void | clear_events () |
| Public Member Functions inherited from Subscriptor | |
| Subscriptor () | |
| Subscriptor (const Subscriptor &) | |
| Subscriptor (Subscriptor &&) noexcept | |
| virtual | ~Subscriptor () |
| Subscriptor & | operator= (const Subscriptor &) |
| Subscriptor & | operator= (Subscriptor &&) noexcept |
| void | subscribe (const char *identifier=nullptr) const |
| void | unsubscribe (const char *identifier=nullptr) const |
| unsigned int | n_subscriptions () const |
| void | list_subscribers () const |
| template<class Archive > | |
| void | serialize (Archive &ar, const unsigned int version) |
Static Public Member Functions | |
| static void | declare_parameters (ParameterHandler ¶m) |
| Static Public Member Functions inherited from Subscriptor | |
| static ::ExceptionBase & | ExcInUse (int arg1, std::string arg2, std::string arg3) |
| static ::ExceptionBase & | ExcNoSubscriber (std::string arg1, std::string arg2) |
Private Attributes | |
| TimestepControl | control |
| double | vtheta |
| bool | adaptive |
| TimestepData | d_explicit |
| TimestepData | d_implicit |
| SmartPointer< OperatorBase, ThetaTimestepping< VectorType > > | op_explicit |
| SmartPointer< OperatorBase, ThetaTimestepping< VectorType > > | op_implicit |
| SmartPointer< OutputOperator< VectorType >, ThetaTimestepping< VectorType > > | output |
Additional Inherited Members | |
| Protected Attributes inherited from Algorithms::OperatorBase | |
| Event | notifications |
Application class performing the theta timestepping scheme.
The theta scheme is an abstraction of implicit and explicit Euler schemes, the Crank-Nicholson scheme and linear combinations of those. The choice of the actual scheme is controlled by the parameter theta as follows.
For fixed theta, the Crank-Nicholson scheme is the only second order scheme. Nevertheless, further stability may be achieved by choosing theta larger than ½, thereby introducing a first order error term. In order to avoid a loss of convergence order, the adaptive theta scheme can be used, where theta=½+c dt.
Assume that we want to solve the equation u' + F(u) = 0 with a step size k. A step of the theta scheme can be written as
\[ M u_{n+1} + \theta k F(u_{n+1}) = M u_n - (1-\theta)k F(u_n). \]
Here, M is the mass matrix. We see, that the right hand side amounts to an explicit Euler step with modified step size in weak form (up to inversion of M). The left hand side corresponds to an implicit Euler step with modified step size (right hand side given). Thus, the implementation of the theta scheme will use two Operator objects, one for the explicit, one for the implicit part. Each of these will use its own TimestepData to account for the modified step sizes (and different times if the problem is not autonomous). Note that once the explicit part has been computed, the left hand side actually constitutes a linear or nonlinear system which has to be solved.
ThetaTimestepping uses AnyData for communicating vectors and time step information. With outer or inner Operator objects. It does not use itself the input vectors provided, but forwards them to the explicit and implicit operators.
The explicit Operator op_explicit receives in its input in first place the vector "Previous iterate", which is the solution value after the previous timestep. It is followed by all vectors provided to ThetaTimestepping::operator() as input argument. op_explicit is supposed to write its result into the first position of its output argument, labeled "Result".
The implicit Operator op_implicit receives the result of op_explicit in its first input vector labeled "Previous time". It is followed by all vectors provided to ThetaTimestepping::operator() as input argument. The output of op_implicit is directly written into the output argument given to ThetaTimestepping.
Since the introduction of AnyData, ThetaTimestepping is able to communicate the current time step information through AnyData as well. Therefore, the AnyData objects handed as input to op_explicit and op_implicit contain two entries of type const double* named "Time" and "Timestep". Note that "Time" refers to the time at the beginning of the current step for op_explicit and at the end for op_implicit, respectively.
The use ThetaTimestepping is more complicated than for instance Newton, since the inner operators will usually need to access the TimeStepData. Thus, we have a circular dependency of information, and we include the following example for its use. It can be found in examples/doxygen/theta_timestepping.cc
First, we define the two operators used by ThetaTimestepping and call them Implicit and Explicit. They both share the public interface of Operator, and additionally provide storage for the matrices to be used and a pointer to TimestepData. Note that we do not use a SmartPointer here, since the TimestepData will be destroyed before the operator.
These operators will be implemented after the main program. But let us look first at how they get used. First, let us define a matrix to be used for our system and also an OutputOperator in order to write the data of each timestep to a file.
Now we create objects for the implicit and explicit parts of the steps as well as the ThetaTimestepping itself. We initialize the timestepping with the output operator in order to be able to see the output in every step.
The next step is providing the vectors to be used. value is filled with the initial value and is also the vector where the solution at each timestep will be. Because the interface of Operator has to be able to handle several vectors, we need to store it in an AnyData object. Since our problem has no additional parameters, the input AnyData object remains empty.
Finally, we are ready to tell the solver, that we are starting at the initial timestep and run it.
First the constructor, which simply copies the system matrix into the member pointer for later use.
Now we need to study the application of the implicit and explicit operator. We assume that the pointer matrix points to the matrix created in the main program (the constructor did this for us). Here, we first get the time step size from the AnyData object that was provided as input. Then, if we are in the first step or if the timestep has changed, we fill the local matrix \(m\), such that with the given matrix \(M\), it becomes
\[ m = I - \Delta t M. \]
After we have worked off the notifications, we clear them, such that the matrix is only generated when necessary.
Now we multiply the input vector with the new matrix and store on output.
that we change the sign in front of the timestep and use the inverse of t he matrix.Definition at line 190 of file theta_timestepping.h.
| Algorithms::ThetaTimestepping< VectorType >::ThetaTimestepping | ( | OperatorBase & | op_explicit, |
| OperatorBase & | op_implicit | ||
| ) |
Constructor, receiving the two operators stored in op_explicit and op_implicit. For their meaning, see the description of those variables.
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virtual |
The timestepping scheme.
| in | is ignored by ThetaTimestepping, but is merged into the AnyData objects used as input for the operators op_explicit and op_implicit. |
| out | in its first argument must contain a pointer to a VectorType instance, which contains the initial value when the operator is called. It contains the final value when the operator returns. |
Implements Algorithms::OperatorBase.
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Register an event triggered by an outer iteration.
Reimplemented from Algorithms::OperatorBase.
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Define an operator which will output the result in each step. Note that no output will be generated without this.
Definition at line 358 of file theta_timestepping.h.
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Declare parameters in a parameter handler.
| void Algorithms::ThetaTimestepping< VectorType >::parse_parameters | ( | ParameterHandler & | param | ) |
Read the parameters in the ParameterHandler.
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The current time in the timestepping scheme.
Definition at line 384 of file theta_timestepping.h.
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The weight between implicit and explicit part.
Definition at line 366 of file theta_timestepping.h.
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Set a new weight and return the old
Definition at line 374 of file theta_timestepping.h.
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The data handed to the op_explicit time stepping operator.
The time in here is the time at the beginning of the current step, the time step is (1-theta) times the actual time step.
Definition at line 333 of file theta_timestepping.h.
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The data handed to the op_implicit time stepping operator.
The time in here is the time at the beginning of the current step, the time step is theta times the actual time step.
Definition at line 342 of file theta_timestepping.h.
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Allow access to the control object.
Definition at line 351 of file theta_timestepping.h.
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The object controlling the time step size and computing the new time in each step.
Definition at line 276 of file theta_timestepping.h.
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The control parameter theta in the range [0,1]. It defaults to 0.5.
Definition at line 282 of file theta_timestepping.h.
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Use adaptive theta if true. Not yet implemented.
Definition at line 286 of file theta_timestepping.h.
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The data for the explicit part of the scheme.
Definition at line 291 of file theta_timestepping.h.
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The data for the implicit part of the scheme.
Definition at line 296 of file theta_timestepping.h.
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The operator computing the explicit part of the scheme. This will receive in its input data the value at the current time with name "Current time solution". It should obtain the current time and time step size from explicit_data().
Its return value is \( Mu+cF(u) \), where \(u\) is the current state vector, \(M\) the mass matrix, \(F\) the operator in space and \(c\) is the adjusted time step size \((1-\theta) \Delta t\).
Definition at line 309 of file theta_timestepping.h.
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The operator solving the implicit part of the scheme. It will receive in its input data the vector "Previous time". Information on the timestep should be obtained from implicit_data().
Its return value is the solution u of Mu-cF(u)=f, where f is the dual space vector found in the "Previous time" entry of the input data, M the mass matrix, F the operator in space and c is the adjusted time step size \( \theta \Delta t\)
Definition at line 321 of file theta_timestepping.h.
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The operator writing the output in each time step
Definition at line 326 of file theta_timestepping.h.
1.8.14