Stability-Preserving Finite-Difference Methods for General Multi-Dimensional Autonomous Dynamical Systems

Abstract

General multi-dimensional autonomous dynamical systems and their numerical discretizations are considered. Nonstandard stability-preserving finite-difference schemes based on the $\theta$-methods and the second-order Runge-Kutta methods are designed and analyzed. Their elementary stability is established theoretically and is also supported by a set of numerical examples.