#include <deal.II/algorithms/theta_timestepping.h>

Inheritance diagram for Algorithms::ThetaTimestepping< VectorType >:

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 &param)

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 () override=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 (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

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Static Public Member Functions
static void	declare_parameters (ParameterHandler &param)

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

Detailed Description

template<typename VectorType>
class Algorithms::ThetaTimestepping< VectorType >

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.

theta=0: explicit Euler scheme
theta=1: implicit Euler scheme
theta=½: Crank-Nicholson scheme

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.

Usage AnyData

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.

Vector data

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.

Scalar data

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.

Usage of ThetaTimestepping

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.

class Explicit : public OperatorBase
{
public:
  Explicit(const FullMatrix<double> &matrix);
  void operator()(AnyData &out, const AnyData &in);
 
private:
  SmartPointer<const FullMatrix<double>, Explicit> matrix;
  FullMatrix<double>                               m;
};
 
class Implicit : public OperatorBase
{
public:
  Implicit(const FullMatrix<double> &matrix);
  void operator()(AnyData &out, const AnyData &in);
 
private:
  SmartPointer<const FullMatrix<double>, Implicit> matrix;
  FullMatrix<double>                               m;
};

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.

int main()
{
  FullMatrix<double> matrix(2);
  matrix(0, 0) = 0.;
  matrix(1, 1) = 0.;
  matrix(0, 1) = 3.14;
  matrix(1, 0) = -3.14;
 
  OutputOperator<Vector<double>> out;
  out.initialize_stream(std::cout);

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.

Explicit                          op_explicit(matrix);
Implicit                          op_implicit(matrix);
ThetaTimestepping<Vector<double>> solver(op_explicit, op_implicit);
solver.set_output(out);

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.

Vector<double> value(2);
value(0) = 1.;
AnyData indata;
AnyData outdata;
outdata.add(&value, "value");

Finally, we are ready to tell the solver, that we are starting at the initial timestep and run it.

  solver.notify(Events::initial);
  solver(outdata, indata);
}

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.

Explicit::Explicit(const FullMatrix<double> &M)
  : matrix(&M)
{
  m.reinit(M.m(), M.n());
}

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.

void Explicit::operator()(AnyData &out, const AnyData &in)
{
  const double timestep = *in.read_ptr<double>("Timestep");
  if (this->notifications.test(Events::initial) ||
      this->notifications.test(Events::new_timestep_size))
    {
      m.equ(-timestep, *matrix);
      for (unsigned int i = 0; i < m.m(); ++i)
        m(i, i) += 1.;
    }
  this->notifications.clear();

Now we multiply the input vector with the new matrix and store on output.

  m.vmult(*out.entry<Vector<double> *>(0),
          *in.read_ptr<Vector<double>>("Previous iterate"));
}

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.

Implicit::Implicit(const FullMatrix<double> &M)
  : matrix(&M)
{
  m.reinit(M.m(), M.n());
}
 
void Implicit::operator()(AnyData &out, const AnyData &in)
{
  const double timestep = *in.read_ptr<double>("Timestep");
  if (this->notifications.test(Events::initial) ||
      this->notifications.test(Events::new_timestep_size))
    {
      m.equ(timestep, *matrix);
      for (unsigned int i = 0; i < m.m(); ++i)
        m(i, i) += 1.;
      m.gauss_jordan();
    }
  this->notifications.clear();
  m.vmult(*out.entry<Vector<double> *>(0),
          *in.read_ptr<Vector<double>>("Previous time"));
}

Author: Guido Kanschat

Date: 2010

Definition at line 289 of file theta_timestepping.h.

Constructor & Destructor Documentation

◆ ThetaTimestepping()

template<typename VectorType >

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.

Member Function Documentation

◆ operator()()

template<typename VectorType >

virtual void Algorithms::ThetaTimestepping< VectorType >::operator()	(	AnyData &	out,
		const AnyData &	in
	)

overridevirtual

The timestepping scheme.

Parameters

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.

◆ notify()

template<typename VectorType >

virtual void Algorithms::ThetaTimestepping< VectorType >::notify ( const Event & )

overridevirtual

Register an event triggered by an outer iteration.

Reimplemented from Algorithms::OperatorBase.

◆ set_output()

template<typename VectorType >

void Algorithms::ThetaTimestepping< VectorType >::set_output ( OutputOperator< VectorType > & output )

inline

Define an operator which will output the result in each step. Note that no output will be generated without this.

Definition at line 465 of file theta_timestepping.h.

◆ declare_parameters()

template<typename VectorType >

static void Algorithms::ThetaTimestepping< VectorType >::declare_parameters ( ParameterHandler & param )

static

template<typename VectorType >

const TimestepData & Algorithms::ThetaTimestepping< VectorType >::explicit_data

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 (1-theta) times the actual time step.

Definition at line 442 of file theta_timestepping.h.

◆ implicit_data()

template<typename VectorType >

const TimestepData & Algorithms::ThetaTimestepping< VectorType >::implicit_data

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 450 of file theta_timestepping.h.

◆ timestep_control()

template<typename VectorType >

template<typename VectorType >

SmartPointer<OperatorBase, ThetaTimestepping<VectorType> > Algorithms::ThetaTimestepping< VectorType >::op_explicit

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 418 of file theta_timestepping.h.

◆ op_implicit

template<typename VectorType >

SmartPointer<OperatorBase, ThetaTimestepping<VectorType> > Algorithms::ThetaTimestepping< VectorType >::op_implicit

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 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 430 of file theta_timestepping.h.

◆ output

template<typename VectorType >

SmartPointer<OutputOperator<VectorType>, ThetaTimestepping<VectorType> > Algorithms::ThetaTimestepping< VectorType >::output

private

The operator writing the output in each time step

Definition at line 436 of file theta_timestepping.h.

The documentation for this class was generated from the following file:

include/deal.II/algorithms/theta_timestepping.h

Public Member Functions

Static Public Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

template<typename VectorType> class Algorithms::ThetaTimestepping< VectorType >

Usage AnyData

Vector data

Scalar data

Usage of ThetaTimestepping

Constructor & Destructor Documentation

◆ ThetaTimestepping()

Member Function Documentation

◆ operator()()

◆ notify()

◆ set_output()

◆ declare_parameters()

◆ parse_parameters()

◆ current_time()

◆ theta() [1/2]

◆ theta() [2/2]

◆ explicit_data()

◆ implicit_data()

◆ timestep_control()

Member Data Documentation

◆ control

◆ vtheta

◆ adaptive

◆ d_explicit

◆ d_implicit

◆ op_explicit

◆ op_implicit

◆ output

template<typename VectorType>
class Algorithms::ThetaTimestepping< VectorType >