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
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!
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
$\theta=1$, 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
+ 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
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
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]
=
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
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
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: <i>Über die partiellen Differenzengleichungen der
-mathematischen Physik</i>, 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: <i>Über die partiellen
+Differenzengleichungen der mathematischen Physik</i>, Mathematische
+Annalen, vol. 100, no. 1, pages 32-74, 1928.)
+This condition on the time step is most frequently just referred
to as the <i>CFL</i> 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