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The wave equation is closely related to the so-called advection equation, which
in one dimension takes the form
![\begin{displaymath}
\frac{\partial u}{\partial t} = -v\,\frac{\partial u}{\partial x}.
\end{displaymath}](img938.png) |
(234) |
This equation describes the passive advection of some scalar field
carried along by a flow of constant speed
. Since the advection equation
is somewhat simpler than the wave equation, we shall discuss it first.
The advection equation
possesses the formal solution
![\begin{displaymath}
u(x,t) = F(x-v\,t),
\end{displaymath}](img940.png) |
(235) |
where
is an arbitrary function. This solution describes an arbitrarily
shaped pulse which is swept along by the flow, at constant speed
, without
changing shape.
We seek the solution of Eq. (234) in the region
,
subject to the simple Dirichlet boundary conditions
.
As usual, we discretize in time on the uniform grid
, for
.
Furthermore, in the
-direction, we discretize on the uniform grid
, for
, where
. Adopting an explicit temporal differencing
scheme, and a centered spatial differencing scheme, Eq. (234) yields
![\begin{displaymath}
\frac{u_i^{n+1} - u_i^n}{\delta t} = - v\,\frac{u_{i+1}^n-u_{i-1}^n}{2\,\delta x},
\end{displaymath}](img942.png) |
(236) |
where
. The above equation can be rewritten
![\begin{displaymath}
u_i^{n+1} = u_i^n -\frac{C}{2}\,(u_{i+1}^n-u_{i-1}^n),
\end{displaymath}](img944.png) |
(237) |
where
.
Let us perform a von Neumann stability analysis of the above differencing scheme.
Writing
, we obtain
, where
![\begin{displaymath}
A = 1 - {\rm i} \,C\,\sin (k \,\delta x).
\end{displaymath}](img948.png) |
(238) |
Note that
![\begin{displaymath}
\vert A\vert^2 = 1 + C^2\,\sin^2(k\,\delta x) > 1.
\end{displaymath}](img949.png) |
(239) |
Thus, the magnitude of the amplification factor is greater than unity for
all
. This implies, unfortunately, that the simple differencing scheme (237)
is unconditionally unstable.
Next: The Lax scheme
Up: The wave equation
Previous: Introduction
Richard Fitzpatrick
2006-03-29