Finite Difference Scheme for the Heat Equation
We apply a finite difference scheme to the heat equation, , and study its convergence. The rate of convergence (or divergence) depends on the problem data and the inhomogeneous function .
Consider the finite difference scheme
, , ,
This Demonstration shows how the convergence of this finite difference scheme depends on the initial data, the boundary values, and the parameter that defines the scheme for the heat equation . If , then the scheme is called explicit; if , it is called implicit. If , then the scheme is stable, so the approximate solution converges to the exact solution. If , the scheme diverges.
The three pairs of snapshots 1–2, 3–4, and 5–6 show the dependence of the convergence on .
 A. A. Samarskii and A. V. Goolin, Numerical Methods (in Russian), Moscow: Science, 1989.