A Trilayer Difference Scheme for One-Dimensional Parabolic Systems

Abstract

In order to obtain the numerical solution for a one-dimensional parabolic system, an unconditionally stable difference method is investigated in [1]. If the number of unknown functions is M, for each time step only M times of calculation are needed. The rate of convergence is $O(\tau+h^2)$. On the basis of [1], an alternating calculation difference scheme is presented in [2]; the rate of the convergence is $O(\tau^2+h^2)$. The difference schemes in [1] and [2] are economic ones. For the $\alpha$-$th$ equation, only $U_{\alpha}$ is an unknown function; the others $U_{\beta}$ are given evaluated either in the last step or in the present step. So the practical calculation is quite convenient.
The purpose of this paper is to derive a trilayer difference scheme for one-dimensional parabolic systems. It is known that the scheme is also unconditionally stable and the rate of convergence is $O(\tau^2+h^2)$.