The Extrapolation of First Order Methods for Parabolic Partial Differential Equations, II

Abstract

In Part I of this paper (SIAM J. Numer. Anal., 15 (1978), pp. 1212–1224), novel algorithms were introduced for solving parabolic differential equations in which high-frequency components occurred in the solution and for which $A_0 $-stable methods, exemplified by the classical Crank-Nicolson method, were less than satisfactory. The algorithms presented were based on a simple extrapolation of the Backward Euler method which produced $L_0 $ -stability. In all cases the algorithms presented were second order accurate in time. In the present paper some of the previous algorithms are generalized to higher orders of accuracy. In particular, a more detailed consideration of the second order algorithm is given which leads naturally to methods of orders three and four. The novel algorithms are tested on a heat equation with constant coefficients in which a discontinuity between the initial values and boundary values exists.