Finite Difference Methods for Parabolic Problems

Abstract

In this chapter we give an introduction to the numerical solution of parabolic equations by finite differences, and consider the application of such methods to the homogeneous heat equation in one space dimension.We first discuss, in Sect. 9.1, the pure initial value problem, with data given on the unrestricted real axis, and then in Sect. 9.2 the mixed initial-boundary value problem on a finite interval in space, under Dirichlet boundary conditions. We discuss stability and error bounds for various choices of finite difference approximations, in maximum-norm by maximum principle type arguments and in L₂-norm by Fourier analysis. For the unrestricted problem we consider explicit schemes, and on a finite interval also implicit ones, such as the Crank-Nicolson scheme.