From: bangerth Date: Mon, 24 Oct 2011 12:24:09 +0000 (+0000) Subject: Link to step-43. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=0a7b12ffa4443500350fd572c979306f6d81947e;p=dealii-svn.git Link to step-43. git-svn-id: https://svn.dealii.org/trunk@24662 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-21/doc/results.dox b/deal.II/examples/step-21/doc/results.dox index 85b68205a9..ad0d87e96b 100644 --- a/deal.II/examples/step-21/doc/results.dox +++ b/deal.II/examples/step-21/doc/results.dox @@ -106,6 +106,10 @@ are not equipped to do this kind of transformation.

Possibilities for extensions

+There are a number of areas where this program can be improved. Three of them +are listed below. All of them are, in fact, addressed in a tutorial program +that forms the continuation of the current one: step-43. +

Solvers

@@ -114,7 +118,7 @@ computation took about a day for the 1,000 or so time steps. The corresponding 3d computation took almost two days for 800 time steps. The reason why it isn't faster than this is twofold. First, we rebuild the entire matrix in every time step, although some parts such as the $B$, $B^T$, and $M^S$ blocks -never change. +never change. Second, we could do a lot better with the solver and preconditioners. Presently, we solve the Schur complement $B^TM^u(S)^{-1}B$ @@ -140,7 +144,7 @@ expensive, but we have to do it only once in the first time step, and can then use it as a cheap preconditioner in the future. We could do better even by using the SparseDirectUMFPACK class that produces not only an incomplete, but a complete decomposition of the matrix, which should yield an even better -preconditioner. +preconditioner. Finally, why use the approximation $B^T (\textrm{diag}(M^u(S)))^{-1} B$ to precondition $B^T M^u(S)^{-1} B$? The latter matrix, after all, is the mixed @@ -158,7 +162,7 @@ will be faster by at least an order of magnitude.

Time stepping

-In the introduction we have identified the time step restriction +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] @@ -186,7 +190,7 @@ 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 +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.