Journal of Computational Mathematics Vol.33, No.1, 2015, 17–32. http://www.global-sci.org/jcm doi:10.4208/jcm.1406-m4443 FINITE DIFFERENCE METHODS FOR THE HEAT EQUATION WITH A NONLOCAL BOUNDARY CONDITION * V. Thom´ ee Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 G¨ oteborg, Sweden E-mail: thomee@chalmers.se A.S. Vasudeva Murthy TIFR Centre for Appl. Math, Yelahanka New Town, Bangalore, India E-mail: vasu@math.tifrbng.res.in 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 θ-method for 0 <θ ≤ 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 θ =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. Mathematics subject classification: 65M06, 65M12, 65M15 Key words: Heat equation, Artificial boundary conditions, unbounded domains, product quadrature. 1. Introduction We are concerned with the numerical solution of the parabolic problem on a semi-infinite interval, u t = u xx + f (x,t), for x ≥ 0, t> 0, (1.1a) u(0,t)= b(t), for t> 0, (1.1b) u(x, 0) = v(x), for x ≥ 0, (1.1c) u → 0, for x → +∞, (1.1d) where f (x,t) and v(x) vanish outside a finite interval in x, which in the sequel we normalize to be [0, 1). To be able to use finite difference or finite element methods for this problem, it is useful to truncate it to this finite spatial interval. This necessitates setting up a boundary condition at the right hand endpoint of the interval, x = 1, usually referred to as an artificial boundary condition (abc). Han and Huang [3] have recently proposed such an abc for (1.1) * Received October 17, 2013 / Revised version received May 19, 2014 / Accepted June 26, 2014 / Published online December 1, 2014 /