From c39d237372686a5d5491d4a05c508342a4e563a2 Mon Sep 17 00:00:00 2001 From: bangerth Date: Thu, 22 May 2008 19:31:18 +0000 Subject: [PATCH] Add possible extensions. git-svn-id: https://svn.dealii.org/trunk@16168 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-33/doc/results.dox | 71 ++++++++++++++++++++++++ 1 file changed, 71 insertions(+) diff --git a/deal.II/examples/step-33/doc/results.dox b/deal.II/examples/step-33/doc/results.dox index c56fa2f649..0ea380133b 100644 --- a/deal.II/examples/step-33/doc/results.dox +++ b/deal.II/examples/step-33/doc/results.dox @@ -152,3 +152,74 @@ We can also visualize the evolution of the adaptively refined grid: The adaptivity follows and preceeds the flow pattern, based on the heuristic refinement scheme discussed above. + + + + +

Possibilities for extensions

+ +

Adaptive time stepping

+ +In the computations shown above, we use a fixed time step. This would +appear sub-optimal: we should take larger steps whenever the problem +is still solvable with Newton's method in 3 or 4 iterations; certainly +more importantly we should take smaller steps whenever we can't solve +the problem at hand any more with Newton's method. The latter is what +happens to the calculation the screen output of which was shown above: +@code +@endcode +If this happens, it would be nice if we could either (i) detect the +problem up front and reduce the time step before we even start the +time step, or (ii) accept the failure at this time step and then +simply start over from the previous time step trying with a reduced +step size. + + +

Stabilization

+ +The numerical scheme we have chosen is not particularly +stable. Furthermore, it is known how to make it more stable, for +example by using streamline diffusion or least-squares stabilization +terms. + + +

Better linear solvers

+ +While the Newton method as a nonlinear solver appears to work very +well if the time step is small enough, the linear solver can be +improved. For example, in the current scheme whenever we use an +iterative solver, an ILU is computed anew for each Newton step; +likewise, for the direct solver, an LU decomposition of the Newton +matrix is computed in each step. This is obviously wasteful: from one +Newton step to another, and probably also between time steps, the +Newton matrix does not radically change: an ILU or a sparse LU +decomposition for one Newton step is probably still a very good +preconditioner for the next Newton or time step. Avoiding the +recomputation would therefore be a good way to reduce the amount of +compute time. + +One could drive this a step further: since close to convergence the +Newton matrix changes only a little bit, one may be able to define a +quasi-Newton scheme where we only re-compute the residual (i.e. the +right hand side vector) in each Newton iteration, and re-use the +Newton matrix. The resulting scheme will likely not be of quadratic +convergence order, and we have to expect to do a few more nonlinear +iterations; however, given that we don't have to spend the time to +build the Newton matrix each time, the resulting scheme may well be +faster. + + +

Other conservation laws

+ +Finally, as a direction beyond the immediate solution of the Euler +equations, this program tries very hard to separate the implementation +of everything that is specific to the Euler equations into one class +(the EulerEquation class), and everything that is +specific to assembling the matrices and vectors, nonlinear and linear +solvers, and the general top-level logic into another (the +ConservationLaw class). + +By replacing the definitions of flux matrices and numerical fluxes in +this class, as well as the various other parts defined there, it +should be possible to apply the ConservationLaw class to +other hyperbolic conservation laws as well. -- 2.39.5