A method of “alternating corrections” for the numerical solution of two-point boundary values problems

Abstract:

In this paper a method of “alternating corrections” is defined and analyzed for the numerical solution of the two-point boundary value problem \begin{align} y” = f(x, y) \\ y(0) = a \\ y(1) = b. \\ \end{align} The case where the first derivative does not enter explicitly into the differential equation is chosen for simplicity of treatment. The alternating corrections method can easily be modified to treat the more general case. The function $f(x,y)$ is assumed to have continuous second derivatives, but the differential equation may, of course, be non-linear. The method to be described is essentially a relaxation technique suitable for an automatic digital computer. The main feature of the method is that most of the “correcting” is done in the early stages of the computation, using a small number of points; thus a rough approximation to the solution is obtained quickly. This approximation can then be made more accurate in the later stages of the computation, as the number of points is increased. In Section 1 the method is described. Section 2 gives a rigorous truncation and stability analysis. Section 3 contains the proof of the convergence of the method giving an estimate of the rate of convergence, and in Section 4 some experimental results obtained on a digital computer are examined.