Scientific Computing

This module illustrates the CFL condition for stability of an
explicit finite difference discretization of the wave equation. The
*domain of dependence* of a hyperbolic partial differential
equation (PDE) for a given point in the problem domain is that portion
of the problem domain that influences the value of the solution at the
given point. Similarly, the domain of dependence of an explicit finite
difference scheme for a given mesh point is the set of mesh points that
affect the value of the approximate solution at the given mesh point.
The *CFL condition*, named for its originators Courant,
Friedrichs, and Lewy, requires that the domain of dependence of the PDE
must lie *within* the domain of dependence of the finite
difference scheme for each mesh point of an explicit finite difference
scheme for a hyperbolic PDE. Any explicit finite difference scheme
that violates the CFL condition is necessarily unstable, but satisfying
the CFL condition does not necessarily guarantee stability. This
module illustrates the CFL condition for the wave equation
*u*_{tt} = *c*
*u*_{xx},*c*
determines the wave speed. For the wave equation, the domain of
dependence for a given point is an isosceles triangle having its apex
at the given point, one edge on the spatial axis, and remaining edges
with slopes *c*^{½}*c*^{½}

The coefficient *c*, spatial step size
*x**t**c*^{½}Δ*t* ⁄Δ*x*
≤ 1

The CFL condition can be interpreted intuitively as requiring that
the distance *c*^{½}Δ*t**x**x* ⁄ Δ*t**c*^{½}*c* is determined by the physical problem being solved,
*x**t* ≤ Δ*x* ⁄
*c*^{½}*t*

**Reference:** Michael T. Heath, *Scientific Computing,
An Introductory Survey*, 2nd edition, McGraw-Hill, New York,
2002. See Section 11.2.2, especially Example 11.4 and Figure 11.6.

**Developers:** Evan VanderZee and Michael Heath