Phase Space Error Control with Variable Time-Stepping Algorithms applied to the Forward Euler method for Dynamical Systems

Abstract

We consider a phase space stability error control for numerical simulation of dynamical systems. Tony Humphries and Vigneswaran, R., in “Phase space stability error control with variable time-stepping Runge-Kutta methods for dynamical systems” illustrated how variable time stepping algorithms performs poorly for long time computations which pass close to a fixed point. A new error control was introduced by Humphries A.R., and Christodoulou.N, in “Phase space error control for dynamical system ii” research report which is generalization of the error control first proposed by D.J. Higham, A.A. Humphries, and R, J. Wain in “Phase space error control for dynamical systems”. For linear systems with a stable hyperbolic fixed point, this error control gives a numerical solution which is forced to converge to the fixed point. In particular, in the above mentioned research of Tony Humphries and Vigneswaran, R., it was analysed only for forward Euler method applied to the linear system whose coefficient matrix has real negative eigenvalues. In this paper we analyze forward Euler method applied to the linear system whose coefficient matrix has complex eigenvalues with negative large real parts. We use same step-size selection scheme which was introduced by Tony Humphries and Vigneswaran, R., in “Phase space stability error control with variable time=stepping Runge-Kutta methods for dynamical systems”. Some theoretical results are obtained and numerical results are given.