|
HyPar
1.0
Finite-Difference Hyperbolic-Parabolic PDE Solver on Cartesian Grids
|
General Linear Methods with Global Error Estimators. More...
#include <string.h>#include <basic.h>#include <arrayfunctions.h>#include <simulation_object.h>#include <timeintegration.h>Go to the source code of this file.
Functions | |
| int | TimeGLMGEE (void *ts) |
General Linear Methods with Global Error Estimators.
Definition in file TimeGLMGEE.c.
| int TimeGLMGEE | ( | void * | ts | ) |
Advance the ODE given by
\begin{equation} \frac{d{\bf u}}{dt} = {\bf F} \left({\bf u}\right) \end{equation}
by one time step of size HyPar::dt using the \(s\)-stage General Linear Method with Global Error Estimation (GLM-GEE), given by
\begin{align} {\bf U}^{\left(i\right)} &= c_{i0}{\bf u}_n + \sum_{j=1}^{r-1} c_{ij} \tilde{\bf u}_n^j + \Delta t \sum_{j=1}^{i-1} a_{ij} {\bf F}\left({\bf U}^{\left(j\right)}\right), i=1,\cdots,s\\ {\bf u}_{n+1} &= d_{00} {\bf u}_n + \sum_{j=1}^{r-1} d_{0j} \tilde{\bf u}_n^j + \Delta t \sum_{j=1}^s b_{0j} {\bf F}\left({\bf U}^{\left(j\right)}\right), \\ \tilde{\bf u}_{n+1}^i &= d_{i0} {\bf u}_n + \sum_{j=1}^{r-1} d_{ij} \tilde{\bf u}_n^j + \Delta t \sum_{j=1}^s b_{ij} {\bf F}\left({\bf U}^{\left(j\right)}\right), i=1,\cdots,r-1 \end{align}
where the superscripts in parentheses represent stages, the subscripts represent the time level, the superscripts without parentheses represent auxiliary solutions, \(\Delta t\) is the time step size HyPar::dt, \(\tilde{\bf u}^i, i=1,\cdots,r-1\) are the auxiliary solutions, and \({\bf F}\left({\bf u}\right)\) is computed by TimeIntegration::RHSFunction. The coefficients defining this methods are:
where \(s\) is the number of stages (GLMGEEParameters::nstages) and \(r\) is the number of auxiliary solutions propagated with the solution (GLMGEEParameters::r).
Note: In the code TimeIntegration::Udot is equivalent to \({\bf F}\left({\bf u}\right)\).
References:
| ts | Object of type TimeIntegration |
Definition at line 45 of file TimeGLMGEE.c.