From 7287ca53b1ea65899bc9d0648f317d50ef7dca6f Mon Sep 17 00:00:00 2001 From: bangerth Date: Wed, 13 Sep 2006 19:15:25 +0000 Subject: [PATCH] Fix a few issues. git-svn-id: https://svn.dealii.org/trunk@13898 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-23/doc/intro.dox | 29 +++++++++++++------------- 1 file changed, 15 insertions(+), 14 deletions(-) diff --git a/deal.II/examples/step-23/doc/intro.dox b/deal.II/examples/step-23/doc/intro.dox index a76f86ebd2..ec995bd1c6 100644 --- a/deal.II/examples/step-23/doc/intro.dox +++ b/deal.II/examples/step-23/doc/intro.dox @@ -37,7 +37,7 @@ Physically, the equation describes the motion of an elastic medium. In 2-d, one can think of how a membrane moves if subjected to a force. The Dirichlet boundary conditions above indicate that the membrane is clamped at the boundary at a height $g(x,t)$ (this height -might be moving as well -- think of people holding a blanket and +might be moving as well — think of people holding a blanket and shaking it up and down). The first initial condition equals the initial deflection of the membrane, whereas the second one gives its velocity. For example, one could think of pushing the membrane down @@ -83,7 +83,7 @@ other time dependent problems, we find that the character of a solution changes as time progresses. For example, for the wave equation, we may have a single wave travelling through the domain, where the solution is smooth or even constant in front of and behind -the wave -- adaptivity would be really useful for such cases, but the +the wave &mdash& adaptivity would be really useful for such cases, but the key is that the area where we need to refine the mesh changes from time step to time step! @@ -96,7 +96,7 @@ standard ODE solvers are certainly not prepared to deal with at all. On the other hand, for the Rothe method, we just get a PDE for each time step that we may choose to discretize independently of the mesh used for the previous time step; this approach is not without -perils and difficulties, but at least it a sensible and well-defined +perils and difficulties, but at least is a sensible and well-defined procedure. For all these reasons, for the present program, we choose to use the @@ -177,7 +177,7 @@ forward or explicit Euler method. On the other hand, if we set $\theta=1$, then we would get $\frac{u^n - u^{n-1}}{k} - v^n = 0$, which corresponds to the backward or implicit Euler method. Both these methods are first order -accurate methods. They are simply to implement, but they are not +accurate methods. They are simple to implement, but they are not really very accurate. The third case would be to choose $\theta=\frac 12$. The first of the @@ -209,7 +209,7 @@ rearranging terms. We then get + k\left[\theta f^n + (1-\theta) f^{n-1}\right]. \f} In this form, we see that if we are given the solution -$u^{n-1},v^{n-1}$ of the previous timestep, that we can the solve for +$u^{n-1},v^{n-1}$ of the previous timestep, that we can then solve for the variables $u^n,v^n$ separately, i.e. one at a time. This is convenient. In addition, we recognize that the operator in the first equation is positive definite, and the second equation looks @@ -259,7 +259,7 @@ V_i^{n-1}\phi_i^{n-1}(x)$. Note that since the solution of the previous time step has already been computed by the time we get to time step $n$, $U^{n-1},V^{n-1}$ are known. Furthermore, note that the solutions of the previous step may have been computed on a different mesh, so -use shape functions $\phi^{n-1}_i(x)$. +we have to use shape functions $\phi^{n-1}_i(x)$. If we plug these expansions into above equations and test with the test functions from the present mesh, we get the following linear @@ -322,7 +322,7 @@ equation. For the wave equation, the natural quantity to look at is the energy. By multiplying the wave equation by $u_t$, integrating over $\Omega$, and integrating by parts where necessary, we find that @f[ - \frac{\partial}{\partial t} + \frac{d}{d t} \left[\frac 12 \int_\Omega \left(\frac{\partial u}{\partial t}\right)^2 + (\nabla u)^2 \; dx\right] = @@ -337,7 +337,8 @@ that E(t) = \frac 12 \int_\Omega \left(\frac{\partial u}{\partial t}\right)^2 + (\nabla u)^2 \; dx @f] -is a conserved quantity. We will compute this quantity after each time +is a conserved quantity, i.e. one that doesn't change with time. We +will compute this quantity after each time step. It is straightforward to see that if we replace $u$ by its finite element approximation, and $\frac{\partial u}{\partial t}$ by the finite element approximation of the velocity $v$, then @@ -358,7 +359,7 @@ explicit time discretizations are only stable if the time step is small enough. In particular, it is coupled to the spatial mesh width $h$. For the lowest order discretization we use here, the relationship reads @f[ - k\le \frac ch + k\le \frac hc @f] where $c$ is the wave speed, which in our formulation of the wave equation has been normalized to one. Consequently, unless we use the implicit schemes with @@ -368,11 +369,11 @@ they become inaccurate if the time step is too large. This condition was first recognized by Courant, Friedrichs, and Lewy — in 1928, long before computers become available for numerical -computations! (This result appeared in R. Courant, K. Friedrichs and -H. Lewy: Über die partiellen Differenzengleichungen der -mathematischen Physik, Mathematische Annalen, vol. 100, no. 1, -pages 32-74, 1928.) -Consequently, the condition on the time step is most frequently just referred +computations! (This result appeared in the German language article +R. Courant, K. Friedrichs and H. Lewy: Über die partiellen +Differenzengleichungen der mathematischen Physik, Mathematische +Annalen, vol. 100, no. 1, pages 32-74, 1928.) +This condition on the time step is most frequently just referred to as the CFL condition. In the program, we will refine a square $[-1,1]$ seven times uniformly, giving a mesh size of $h=\frac 1{64}$, which is what we set the time step to. The fact that we set the time step and mesh -- 2.39.5