Asymptotic stability analysis of the linear theta-method for linear parabolic differential equations with delay

摘要

This paper is concerned with asymptotic stability property of linear theta-method for partial functional differential equations with delay. A sufficient condition for the underlying partial functional differential equations to be asymptotically stable is presented. We investigate numerical stability of the linear theta-method by using the spectral radius condition. When theta is an element of [0, 1/2), a sufficient and necessary condition for the linear theta-method to be asymptotically stable is established. When theta is an element of [1/2, 1], the linear theta-method is unconditionally asymptotically stable. The behaviour of the norm of the iteration matrix when the linear theta-method is asymptotically stable is studied by using Kreiss resolvent condition. Numerical experiments have been implemented to confirm the derived stability properties of the numerical method.