The box scheme proposed by H. B. Keller is a numericalmethod for solving parabolic partial differential equations. We give a convergence proof of this scheme for the heat equation, for a linear parabolic system, and for a class of nonlinear parabolic equations. Von Neumann stability is shown to hold for the box scheme combinedwith the method of fractional steps to solve the two-dimensional heat equation. Computations were performed on Burgers' equation with three different initial conditions, and Richardson extrapolation is shown to be effective.
