The splitting finite-difference scheme for two-dimensional heat conduction equation with four nonlocal integral conditions

Abstract

We construct and analyse a splitting finite-difference scheme for a two-dimensional heat conduction (parabolic) equation with nonlocal integral conditions which are formulated instead of classical boundary conditions. Nonlocal conditions involve integral terms with constant weights. The constructed finite-difference scheme is weighted. With particular values of the weight we have locally one-dimensional (LOD), alternating direction implicit (ADI) or fully-explicit splitting schemes. 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. To be precise, the main point of the analysis of the stability of the scheme is to investigate the spectral structure of two particular quasi-tridiagonal matrices and to verify whether these matrices are simple-structured matrices with real positive eigenvalues. In order to demonstrate the efficiency of the considered finite-difference schemes and practically justify the stability analysis technique, a particular test problem has been solved. The results of numerical experiments with a test problem are presented and they validate theoretical results.