Dumux::Experimental::MultiStageMethod< Scalar > Class Template Reference

Abstract interface for one-step multi-stage method parameters in Shu/Osher form. More...

#include <dumux/timestepping/multistagemethods.hh>

Inheritance diagram for Dumux::Experimental::MultiStageMethod< Scalar >:

Description

template<class Scalar>
class Dumux::Experimental::MultiStageMethod< Scalar >

Abstract interface for one-step multi-stage method parameters in Shu/Osher form.

forward declaration

This implementation is based on the Shu/Osher form from: Chi W. Shu and Stanley Osher. Efficient implementation of essentially non- oscillatory shock-capturing schemes. J. Comput. Phys., 77:439–471, 1988. https://doi.org/10.1016/0021-9991(88)90177-5. To this end Eq. (2.12) is extended for implicit schemes.

We consider the general PDE form

$$\begin{equation} \frac{\partial M(x)}{\partial t} - R(x, t) = 0, \end{equation}$$

where $M(x)$ is the temporal operator/residual in terms of the solution $x$, and $R(x)$ is the discrete spatial operator/residual. $M(x)$ usually corresponds to the conserved quantity (e.g. mass), and $R(x)$ contains the rest of the residual. We can then construct $m$-stage time discretization methods. Integrating from time $t^n$ to $t^{n+1}$ with time step size $\Delta t^n$, we solve:

$$\begin{aligned} x^{(0)} &= u^n\\ \sum_{k=0}^i \left[ \alpha_{ik} M\left(x^{(k)}, t^n + d_k\Delta t^n\right) + \beta_{ik}\Delta t^n R \left(x^{(k)}, t^n+d_k\Delta t^n \right)\right] &= 0 & \forall i \in \{1,\ldots,m\} \\ x^{n+1} &= x^{(m)} \end{aligned}$$

where $x^{(k)}$ denotes the intermediate solution at stage $k$. Dependent on the number of stages $m$, and the coefficients $\alpha, \beta, d$, schemes with different properties and order of accuracy can be constructed.

Public Member Functions
virtual bool	implicit () const =0
virtual std::size_t	numStages () const =0
virtual Scalar	temporalWeight (std::size_t i, std::size_t k) const =0
	weights of the temporal operator residual ( $\alpha_{ik}$) More...
virtual Scalar	spatialWeight (std::size_t i, std::size_t k) const =0
	weights of the spatial operator residual ( $\beta_{ik}$) More...
virtual Scalar	timeStepWeight (std::size_t k) const =0
	time step weights for each stage ( $d_k$) More...
virtual std::string	name () const =0
virtual	~MultiStageMethod ()=default

Constructor & Destructor Documentation

~MultiStageMethod()

template<class Scalar >

virtual Dumux::Experimental::MultiStageMethod< Scalar >::~MultiStageMethod ( )

virtualdefault

Member Function Documentation

implicit()

template<class Scalar >

virtual bool Dumux::Experimental::MultiStageMethod< Scalar >::implicit ( ) const

pure virtual

Implemented in Dumux::Experimental::MultiStage::Theta< Scalar >, and Dumux::Experimental::MultiStage::RungeKuttaExplicitFourthOrder< Scalar >.

name()

template<class Scalar >

virtual std::string Dumux::Experimental::MultiStageMethod< Scalar >::name ( ) const

pure virtual

Implemented in Dumux::Experimental::MultiStage::ExplicitEuler< Scalar >, Dumux::Experimental::MultiStage::ImplicitEuler< Scalar >, Dumux::Experimental::MultiStage::RungeKuttaExplicitFourthOrder< Scalar >, and Dumux::Experimental::MultiStage::Theta< Scalar >.

numStages()

template<class Scalar >

virtual std::size_t Dumux::Experimental::MultiStageMethod< Scalar >::numStages ( ) const

pure virtual

Implemented in Dumux::Experimental::MultiStage::Theta< Scalar >, and Dumux::Experimental::MultiStage::RungeKuttaExplicitFourthOrder< Scalar >.

spatialWeight()

template<class Scalar >

virtual Scalar Dumux::Experimental::MultiStageMethod< Scalar >::spatialWeight ( std::size_t  i, std::size_t  k  ) const

pure virtual

weights of the spatial operator residual ( $\beta_{ik}$)

Implemented in Dumux::Experimental::MultiStage::RungeKuttaExplicitFourthOrder< Scalar >, and Dumux::Experimental::MultiStage::Theta< Scalar >.

temporalWeight()

template<class Scalar >

virtual Scalar Dumux::Experimental::MultiStageMethod< Scalar >::temporalWeight ( std::size_t  i, std::size_t  k  ) const

pure virtual

weights of the temporal operator residual ( $\alpha_{ik}$)

Implemented in Dumux::Experimental::MultiStage::RungeKuttaExplicitFourthOrder< Scalar >, and Dumux::Experimental::MultiStage::Theta< Scalar >.

timeStepWeight()

template<class Scalar >

virtual Scalar Dumux::Experimental::MultiStageMethod< Scalar >::timeStepWeight ( std::size_t  k ) const

pure virtual

time step weights for each stage ( $d_k$)

Implemented in Dumux::Experimental::MultiStage::Theta< Scalar >, and Dumux::Experimental::MultiStage::RungeKuttaExplicitFourthOrder< Scalar >.

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

• multistagemethods.hh