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

(235) |

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

where .

Let us perform a von Neumann stability analysis of the above differencing scheme.
Writing
, we obtain
, where

(238) |

(239) |