Stability Criteria for Finite Difference Approximations to Parabolic Systems—An Update

Abstract

In a recent series of papers, Goldberg [G1, G2] and Sun and Yuan [SY] studied the L ₂-stability of a well-known family of finite difference approximations for the initial-value problem associated with the multispace-dimensional parabolic system

$$\frac{{\partial {\text{u(x,}}\;t{\text{)}}}}{{\partial t}} = \sum\limits_{1 \leqslant {\kern 1pt} p{\kern 1pt} \leqslant {\kern 1pt} q{\kern 1pt} \leqslant {\kern 1pt} {\kern 1pt} s} {A_{pq} \frac{{\partial ^2 {\text{u(x,}}\;t{\text{)}}}}{{\partial x_p \partial x_q }}} + \sum\limits_{1 \leqslant {\kern 1pt} p{\kern 1pt} \leqslant {\kern 1pt} s} {B_p \frac{{\partial {\text{u(x,}}\;t{\text{)}}}}{{\partial x_p }} + C{\text{u(x,}}\;t{\text{)}}} $$

where A _pq ,B _p and C are constant matrices, A _pq being Hermitian. In the present paper we discuss these earlier results and complete the underlying theory by answering four open questions.