The Extrapolation of First Order Methods for Parabolic Partial Differential Equations. I

Abstract

Splitting methods for parabolic partial differential equations based on $(1,1)$ Padé approximations (Crank–Nicolson replacements) are well known to produce poor numerical results when a time discretization is imposed with time steps which are "too large" relative to the spatial discretization. In particular such numerical solutions exhibit an oscillatory behaviour which increases in amplitude with a reduction in the spatial discretization, keeping the time step constant. In contrast, $(1,0)$ Padé approximations (backward-difference, or fully implicit replacements) do not suffer from these drawbacks but are of lower order accuracy. In the present paper, a combination of fully implicit methods is used to attain second order accuracy and to retain the favorable property of the fully implicit scheme. The method is tested on a heat equation in two space dimensions which possesses a discontinuity between the initial and boundary conditions.