A Note on Trapezoidal Methods for the Solution of Initial Value Problems

The trapezoidal rule for the numerical integration of first-order ordinary differential equations is shown to possess, for a certain type of problem, an undesirable property. The removal of this difficulty is shown to be straightforward, resulting in a modified trapezoidal rule. Whilst this latent difficulty is slight (and probably rare in practice), the fact that the proposed modification involves negligible additional programming effort would suggest that it is worthwhile. A corresponding modification for the trapezoidal rule for the Goursat problem is also included. 1. The Trapezoidal Rule. We consider the numerical solution of the initial value problem (1.1) y' = f(x, y), y(x„) = y„ in the region x0 = x = X. The trapezoidal rule for the numerical solution of (1.1) is given by (1.2) y„+i y„ = [f(xB+1, yn+1) + f(x„, y„)], where xn+i = xn + h, h being the mesh length in the x direction. Equation (1.2) is a one-step implicit finite-difference method which is frequently employed for the numerical solution of (1.1). In fact, it is well known that it is the most accurate ^4-stabIe multistep method [1]. A method M is said to be yi-stable if all solutions of M tend to zero as n —* °° when the method is applied with fixed positive h to any differential equation of the form (1.3) y' = -\y, where X is a complex constant with positive real part. For such an equation, (1.2) can be seen to reduce to y«+i i + fx The purpose of this note is simply to point out that there are certain problems, only slightly more general than (1.3), whose numerical solution by (1.2) is totally unaccepReceived October 24, 1969, revised November 26, 1969. AMS Subject Classifications. Primary 6560, 6561, 6567; Secondary 6555.