Numerical method for solving an initial-boundary value problem for a multidimensional loaded parabolic equation of a general form with conditions of the third kind

Abstract

An initial-boundary value problem is studied for a multidimensional loaded parabolic equation of general form with boundary conditions of the third kind. For a numerical solution, a locally one-dimensional difference scheme by A.A. Samarskii with order of approximation O(h2+t) is constructed. Using the method of energy inequalities, we obtain a priori estimates in the differential and difference interpretations, which imply unique- ness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem in the L2 norm at a rate equal to the order of approximation of the difference scheme. An algorithm for the computational solution is constructed and numerical calculations of test cases are carried out, illustrating the theoretical calculations obtained in this work.