Reference documentation for deal.II version Git 8ba89c874e 20210804 22:06:33 0600

#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) override 
virtual void  notify (const Event &) override 
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 () 
void  clear_events () 
template<class Archive >  
void  serialize (Archive &ar, const unsigned int version) 
Subscriptor functionality  
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.  
void  subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const 
void  unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const 
unsigned int  n_subscriptions () const 
template<typename StreamType >  
void  list_subscribers (StreamType &stream) const 
void  list_subscribers () const 
Static Public Member Functions  
static void  declare_parameters (ParameterHandler ¶m) 
static ::ExceptionBase &  ExcInUse (int arg1, std::string arg2, std::string arg3) 
static ::ExceptionBase &  ExcNoSubscriber (std::string arg1, std::string arg2) 
Protected Attributes  
Event  notifications 
Application class performing the theta timestepping scheme.
The theta scheme is an abstraction of implicit and explicit Euler schemes, the CrankNicholson scheme and linear combinations of those. The choice of the actual scheme is controlled by the parameter theta as follows.
For fixed theta, the CrankNicholson 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.
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.
Besides the main function, we need to define the members functions of the implicit and explicit operators. 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.
The code for the implicit operator is almost the same, except that we change the sign in front of the timestep and use the inverse of the matrix.
Definition at line 287 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.

overridevirtual 
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.

overridevirtual 
Register an event triggered by an outer iteration.
Reimplemented from Algorithms::OperatorBase.

inline 
Define an operator which will output the result in each step. Note that no output will be generated without this.
Definition at line 463 of file theta_timestepping.h.

static 
Declare parameters in a parameter handler.
void Algorithms::ThetaTimestepping< VectorType >::parse_parameters  (  ParameterHandler &  param  ) 
Read the parameters in the ParameterHandler.

inline 
The current time in the timestepping scheme.
Definition at line 489 of file theta_timestepping.h.

inline 
The weight between implicit and explicit part.
Definition at line 471 of file theta_timestepping.h.

inline 
Set a new weight and return the old
Definition at line 479 of file theta_timestepping.h.

inline 
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 (1theta) times the actual time step.
Definition at line 440 of file theta_timestepping.h.

inline 
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 448 of file theta_timestepping.h.

inline 
Allow access to the control object.
Definition at line 456 of file theta_timestepping.h.

inherited 
Clear all notifications.
Definition at line 49 of file operator.cc.

inherited 
Subscribes a user of the object by storing the pointer validity
. The subscriber may be identified by text supplied as identifier
.
Definition at line 136 of file subscriptor.cc.

inherited 
Unsubscribes a user from the object.
identifier
and the validity
pointer must be the same as the one supplied to subscribe(). Definition at line 156 of file subscriptor.cc.

inlineinherited 
Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.
Definition at line 301 of file subscriptor.h.

inlineinherited 
List the subscribers to the input stream
.
Definition at line 318 of file subscriptor.h.

inherited 
List the subscribers to deallog
.
Definition at line 204 of file subscriptor.cc.

inlineinherited 
Read or write the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.
This function does not actually serialize any of the member variables of this class. The reason is that what this class stores is only who subscribes to this object, but who does so at the time of storing the contents of this object does not necessarily have anything to do with who subscribes to the object when it is restored. Consequently, we do not want to overwrite the subscribers at the time of restoring, and then there is no reason to write the subscribers out in the first place.
Definition at line 310 of file subscriptor.h.

private 
The object controlling the time step size and computing the new time in each step.
Definition at line 383 of file theta_timestepping.h.

private 
The control parameter theta in the range [0,1]
. It defaults to 0.5.
Definition at line 389 of file theta_timestepping.h.

private 
Use adaptive theta if true
. Not yet implemented.
Definition at line 393 of file theta_timestepping.h.

private 
The data for the explicit part of the scheme.
Definition at line 398 of file theta_timestepping.h.

private 
The data for the implicit part of the scheme.
Definition at line 403 of file theta_timestepping.h.

private 
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 416 of file theta_timestepping.h.

private 
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 MucF(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 428 of file theta_timestepping.h.

private 
The operator writing the output in each time step
Definition at line 434 of file theta_timestepping.h.

protectedinherited 
Accumulate events here. If any of those is set, the function solve() of a terminal application must take care of reassembling the matrix.
Definition at line 97 of file operator.h.