Convergence of a difference scheme for the heat equation in a long strip by artificial boundary conditions

The numerical solution of the heat equation on a strip in two dimensions is considered. An artificial boundary is introduced to make the computational domain finite. On the artificial boundary, an exact boundary condition is proposed to reduce the original problem to an initial-boundary value problem in a finite computational domain, A difference scheme is constructed by the method of reduction of order to solve the problem in the finite computational domain. It is proved that the difference scheme is uniquely solvable. unconditionally stable and convergent with the convergence order 2 in space and order 3/2 in time in an energy norm. A numerical example demonstrates the theoretical results.