Stability of θ-schemes for partial differential equations with piecewise constant arguments of alternately retarded and advanced type

ABSTRACT

This paper deals with the analytical and numerical stability of a partial differential equation with piecewise constant arguments of alternately retarded and advanced type. Firstly, the theory of separation of variables in matrix form and the Fourier method are implemented to achieve the sufficient conditions under which the analytic solution is asymptotically stable. Secondly, the discrete equation is obtained by applying the θ-schemes to the original continuous equation, the sufficient conditions for the asymptotic stability of numerical solution are also shown when the mesh ratio satisfying certain conditions. Finally, some numerical experiments for verifying the theoretical results are provided.