Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions

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