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3.4
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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3.4
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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Abstract interface for one-step multi-stage method parameters in Shu/Osher form. More...
#include <dumux/timestepping/multistagemethods.hh>
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 quanity (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 |
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virtualdefault |
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pure virtual |
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pure virtual |
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pure virtual |
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pure virtual |
weights of the spatial operator residual ( \( \beta_{ik} \))
Implemented in Dumux::Experimental::MultiStage::RungeKuttaExplicitFourthOrder< Scalar >, and Dumux::Experimental::MultiStage::Theta< Scalar >.
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pure virtual |
weights of the temporal operator residual ( \( \alpha_{ik} \))
Implemented in Dumux::Experimental::MultiStage::RungeKuttaExplicitFourthOrder< Scalar >, and Dumux::Experimental::MultiStage::Theta< Scalar >.
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pure virtual |
time step weights for each stage ( \( d_k \))
Implemented in Dumux::Experimental::MultiStage::Theta< Scalar >, and Dumux::Experimental::MultiStage::RungeKuttaExplicitFourthOrder< Scalar >.
1.9.3