From 7498f1d81e5e2806f7f5dceaed583b8efba36ed2 Mon Sep 17 00:00:00 2001 From: bangerth Date: Wed, 8 Feb 2012 21:24:39 +0000 Subject: [PATCH] Add a remark. git-svn-id: https://svn.dealii.org/trunk@25016 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-21/doc/intro.dox | 32 +++++++++++++++++++++++++- 1 file changed, 31 insertions(+), 1 deletion(-) diff --git a/deal.II/examples/step-21/doc/intro.dox b/deal.II/examples/step-21/doc/intro.dox index 234958a49d..15188b6e17 100644 --- a/deal.II/examples/step-21/doc/intro.dox +++ b/deal.II/examples/step-21/doc/intro.dox @@ -459,7 +459,37 @@ water is given by F(S)=\frac{S^2}{S^2+\mu (1-S)^2} @f] -Finally, we will show results for computations with the two permeability +@note Coming back to this testcase in step-43 several years later revealed an +oddity in the setup of this testcase. To this end, consider that we can +rewrite the advection equation for the saturation as $S_{t} + (\mathbf{u} +F'(S)) \cdot \nabla S = 0$. Now, at the initial time, we have $S=0$, and with +the given choice of function $F(S)$, we happen to have $F'(0)=0$. In other +words, at $t=0$, the equation reduces to $S_t=0$ for all $\mathbf x$, so the +saturation is zero everywhere and it is going to stay zero everywhere! This is +despite the fact that $\mathbf u$ is not necessarily zero: the combined fluid +is moving, but we've chosen our partial flux $F(S)$ in such a way that +infinitesimal amounts of wetting fluid also only move at infinitesimal speeds +(i.e., they stick to the medium more than the non-wetting phase in which they +are embedded). That said, how can we square this with the knowledge that +wetting fluid is invading from the left, leading to the flow patterns seen in +the results section? That's where we get into +mathematics: Equations like the transport equation we are considering here +have infinitely many solutions, but only one of them is physical: the one that +results from the so-called viscosity limit, called the viscosity +solution. The thing is that with discontinuous elements we arrive at this +viscosity limit because using a numerical flux introduces a finite amount of +artificial viscosity into the numerical scheme. On the other hand, in step-43, +we use an artificial viscosity that is proportional to $\|\mathbf u F'(S)\|$ +on every cell, which at the initial time is zero. Thus, the saturation there is +zero and remains zero; the solution we then get is one solution of the +advection equation, but the method does not converge to the viscosity solution +without further changes. We will therefore use a different initial condition in +that program. + + +Finally, to come back to the description of the testcase, we will show results +for computations with the two permeability functions introduced at the end of the results section of @ref step_20 "step-20":