A Theorem for the Stability of General Predictor-Corrector Methods for the Solution of Systems of Differential Equations

Abstract

Practically, predictor-corrector methods for the solution of differential equations are widely used. In these numerical methods, the question of stability is most important. For a single differential equation Crane and Lambert (1962) studied the stability of a fourth-order generalized corrector formula; Chase (1962) and subsequently Emanuel (1963) demonstrated for particular methods that the overall integration procedure does not generally possess the same stability properties as the corrector except when the step of integration tends to zero. In the present paper it is shown that it is possible to develop a necessary and sufficient condition for the stability of general predictor-corrector methods for the solution of systems of differential equations.KEY WORDS AND PHRASES: stability, predictor-corrector, system of differential equations, Hermitian form, eigenvalues, Jacobian matrix, system of differential equations CR CATEGORIES: 5.17The system of m ordinary differential equations can be written in vector form as follows:with the initial value y' ~-~ -= f(x; y), x0, xe] and the column vectorsDefine the matrices Y = (y.+¢) and F = (f,+j), j = -q ( 1 ) 0 , and the column vectors a = (aj), b -~ (be), a = (dj), g = (~j), j = --q(1)0.THEOREM. The general predictor-corrector method P.+i = Ya "4-hFb, m,~+l = Pn+l "4-am(p,, --on),(2) cn+~ = Y£ + hFf~ + hblf(x,~+l ; ms+l), y.+1 = c,~+I "4-a~(p~+l -ca+i), (ay -am = 1),