Stability of the weighted splitting finite-difference scheme for a two-dimensional parabolic equation with two nonlocal integral conditions

Abstract

Nonlocal conditions arise in mathematical models of various physical, chemical or biological processes. Therefore, interest in developing computational techniques for the numerical solution of partial differential equations (PDEs) with various types of nonlocal conditions has been growing fast. We construct and analyse a weighted splitting finite-difference scheme for a two-dimensional parabolic equation with nonlocal integral conditions. The main attention is paid to the stability of the method. We apply the stability analysis technique which is based on the investigation of the spectral structure of the transition matrix of a finite-difference scheme. We demonstrate that depending on the parameters of the finite-difference scheme and nonlocal conditions the proposed method can be stable or unstable. The results of numerical experiments with several test problems are also presented and they validate theoretical results.