Finite Difference Methods for the Heat Equation with a Nonlocal Boundary Condition

Abstract

We consider the numerical solution by finite difference methods of the heat equation in one space dimension, with a nonlocal integral boundary condition, resulting from the truncation to a finite interval of the problem on a semi-infinite interval. We first analyze the forward Euler method, and then the theta-method for 0 < theta <= 1, in both cases in maximum-norm, showing O(h(2) + k) error bounds, where h is the mesh-width and k the time step. We then give an alternative analysis for the case theta = 1/2, the Crank-Nicolson method, using energy arguments, yielding a O(h(2) + k(3/2)) error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.