A difference scheme for Burgers equation in an unbounded domain

The numerical solution of the one-dimensional Burgers equation in an unbounded domain is considered. Two artificial boundaries are introduced to make the computational domain finite. On both artificial boundaries, two exact boundary conditions are proposed, respectively, 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. At each time level, only a strictly diagonal dominated tridiagonal system of linear algebraic equations needs to be solved. It is proved that the difference scheme is uniquely solvable and unconditional convergent with the convergence order 3/2 in time and order 2 in space in an energy norm. A numerical example demonstrates the theoretical results.