From: bangerth Date: Mon, 9 Jun 2008 22:12:59 +0000 (+0000) Subject: State other possible avenues for future studies. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=08cd534e4c868c1e3450915c18755137ec52bbcb;p=dealii-svn.git State other possible avenues for future studies. git-svn-id: https://svn.dealii.org/trunk@16283 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-21/doc/intro.dox b/deal.II/examples/step-21/doc/intro.dox index 07e2a4651c..3d5d70a181 100644 --- a/deal.II/examples/step-21/doc/intro.dox +++ b/deal.II/examples/step-21/doc/intro.dox @@ -385,7 +385,9 @@ anything useful once we have negative saturations or ones larger than one. Note that we will have similar restrictions on the time step also in @ref step_23 "step-23" and @ref step_24 "step-24" where we solve the time dependent -wave equation, another hyperbolic problem. +wave equation, another hyperbolic problem. We will also come back to the issue +of time step choice below in the section on possible +extensions to this program.

The test case

diff --git a/deal.II/examples/step-21/doc/results.dox b/deal.II/examples/step-21/doc/results.dox index 99f28c148a..8e339da8b6 100644 --- a/deal.II/examples/step-21/doc/results.dox +++ b/deal.II/examples/step-21/doc/results.dox @@ -156,6 +156,41 @@ Using such techniques, it can reasonably be expected that the solution process will be faster by at least an order of magnitude. +

Time stepping

+ +In the introduction we have identified the time step restriction +@f[ + \triangle t_{n+1} \le \frac h{|\mathbf{u}^{n+1}(\mathbf{x})|} +@f] +that has to hold globally, i.e. for all $\mathbf x$. After discretization, we +satisfy it by choosing +@f[ + \triangle t_{n+1} = \frac {\min_K h_K}{\max_{\mathbf{x}}|\mathbf{u}^{n+1}(\mathbf{x})|}. +@f] + +This restriction on the time step is somewhat annoying: the finer we make the +mesh the smaller the time step; in other words, we get punished twice: each +time step is more expensive to solve and we have to do more time steps. + +This is particularly annoying since the majority of the additional work is +spent solving the implicit part of the equations, i.e. the pressure-velocity +system, whereas it is the hyperbolic transport equation for the saturation +that imposes the time step restriction. + +To avoid this bottleneck, people have invented a number of approaches. For +example, they may only re-compute the pressure-velocity field every few time +steps (or, if you want, use different time step sizes for the +pressure/velocity and saturation equations). This keeps the time step +restriction on the cheap explicit part while it makes the solution of the +implicit part less frequent. Experiments in this direction are +certainly worthwhile; one starting point for such an approach is the paper by +Zhangxin Chen, Guanren Huan and Baoyan Li: An improved IMPES method for +two-phase flow in porous media, Transport in Porous Media, 54 (2004), +pp. 361--376. There are certainly many other papers on this topic as well, but +this one happened to land on our desk a while back. + + +

Adaptivity

Adaptivity would also clearly help. Looking at the movies, one clearly sees