From ad25de4641e2e939ac162dc7a89abe009b779865 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Fri, 21 Aug 2009 17:20:39 +0000 Subject: [PATCH] Write a couple of paragraphs. git-svn-id: https://svn.dealii.org/trunk@19330 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-32/doc/intro.dox | 87 ++++++++++++++++++++++++++ 1 file changed, 87 insertions(+) diff --git a/deal.II/examples/step-32/doc/intro.dox b/deal.II/examples/step-32/doc/intro.dox index e5e7812a80..83b0413b9c 100644 --- a/deal.II/examples/step-32/doc/intro.dox +++ b/deal.II/examples/step-32/doc/intro.dox @@ -91,6 +91,93 @@ section of the @ref step_22 "step-22" tutorial program, where we observed that BiCGStab does not like inner solves with CG, which made us prefer GMRES in step-31. + +

Changes to the artificial viscosity stabilization

+ +As in @ref step_31 "step-31", we will use an artificial viscosity of +the form +@f{eqnarray*} + \nu_\alpha(T)|_K + = + \beta + \|\mathbf{u}\|_{L^\infty(K)} + \min\left\{ + h_K, + h_K^\alpha + \frac{\|R_\alpha(T)\|_{L^\infty(K)}}{c(\mathbf{u},T)} + \right\} +@f} +in this problem, where +$c(\mathbf{u},T) = + c_R\ \|\mathbf{u}\|_{L^\infty(\Omega)} \ \mathrm{var}(T) + \ |\mathrm{diam}(\Omega)|^{\alpha-2}$ (for the meaning of the various +terms in these formulas, see @ref step_31 "step-31". In the results +section of that program, we have discussed our choice for $c_R$ and +how we arrived at the value used there mostly by accident, and in more +detail how $\beta$ was chosen. For the +current program, we want to go about this issue a bit more +systematically for both parameters, +using the same line of reasoning with which we chose two other +parameters in our discretization, $c_k$ and $\beta$, in the results +section of step-31. In particular, remember that we would like to make +the artificial viscosity as small as possible while keeping it as large as +necessary. To see what is happening, note that below we will impose +boundary conditions for the temperature between 973 and 4273 Kelvin, +and initial conditions are also chosen in this range; because there +are no internal heat sources or sinks, the temperature should +consequently always be in this range, barring any internal +oscillations. If the minimal temperature drops below 973 Kelvin, then +we need to add stabilization by either increasing $\beta$ or +decreasing $c_R$. + +As we did in step-31, we first determine an optimal value of $\beta$ +by using the "traditional" formula +@f{eqnarray*} + \nu_\alpha(T)|_K + = + \beta + \|\mathbf{u}\|_{L^\infty(K)} + h_K, +@f} +which we know to be stable if only $\beta$ is large enough. Doing a +couple hundred time steps (on a coarser mesh than the one shown in the +program, and with a different viscosity that affects transport +velocities and therefore time step sizes) in 2d will produce the +following graph: + +@image html step-32.beta.2d.png + +As can be seen, values $\beta \le 0.05$ are too small whereas +$\beta=0.052$ appears to work, at least to the time horizon shown +here. As a remark on the side, there are at least two questions one +may wonder here: First, what happens at the time when the solution +becomes unstable? Looking at the graphical output, we can see that +with the unreasonably coarse mesh chosen for these experiments, around +time $t=10^{15}$ seconds the plumes of hot material that have been +rising towards the cold outer boundary and have then spread sideways +are starting to get close to each other, squezzing out the cold +material inbetween. This creates a layer of cells into which fluids +flows from two opposite sides and flows out toward a third, apparently +a scenario that then produce these instabilities without sufficient +stabilization. Second: In step-31, we used +$\beta=0.015\cdot\text{dim}$; why does this not work here? The answer +to this is not entirely clear -- stabilization parameters are +certainly known to depend on things like the shape of cells, for which +we had square in step-31 but have trapezoids in the current +program. Whatever the exact cause, we at least have a value of +$\beta$, namely 0.052 for 2d, that works for the current program. + +With this value fixed, we can go back to the original formula for the +viscosity $\nu$ and play with the constant $c_R$, making it as large +as possible in order to make $\nu$ as small as possible. This gives us +a picture like this: + +@image html doc/step-32.beta_cr.2d.png + +Consequently, $c_R=0.1$ would appear to be the right value. + + +

Parallelization on clusters

Parallelization of scientific codes across multiple machines in a cluster of -- 2.39.5