On the convergence of approximate solutions of the heat equation to the exact solution

Introduction. In many cases it is practically impossible to solve an initial value problem for a partial differential equation exactly although it can be proved that the exact solution does exist and is uniquely determined. Therefore the partial differential equation is very often replaced by a difference equation which is easier to solve and which furnishes an approximation to the solution of the original problem. Three questions arise immediately. (a) If x and t are the independent variables, does the mesh ratio r (r=At/(Ax)2 in the parabolic case) have any influence on the convergence or stability of the approximate solution? (b) Which one of the difference equations leading to the differential equation will furnish a good approximation? (c) How do the initial values of the problem for the difference equation have to be chosen in order to furnish a good approximation? The author believes that the first question has been overemphasized while the importance of the second and third has not been fully realized. This is primarily due to the fact that, in the paper by O'Brien, Hyman, and Kaplan []1 where von Neumann's test of stability is introduced into the literature, it is erroneously stated that a positive answer to von Neumann's test is necessary and sufficient for convergence. As an example it is pointed out in their paper that the numerical solution of Richardson [2] for the problem of finding the temperature in a slab with faces at temperature zero does not converge to the exact solution because von Neumann's test shows instability for all r> 0.2