Abstract: Based on the ideas of the polynomial fitting numerical boundary scheme (SFEBS) and the Taylor expansion boundary scheme (TEBS), a fourth-order numerical boundary scheme (SF-TEBS4) is proposed in this paper. The SF-TEBS4 is an extension of the optimized fourth-order staggered tridiagonal compact difference scheme (OCS4) and the corresponding interpolation scheme (OCI4) on the staggered grid system, developed by the authors recently, for the equations with non-periodical physical boundary conditions. The asymptotic stability of the overall difference scheme, the combination of the numerical boundary scheme SF-TEBS4, and the inner points schemes OCS4 and OCI4, is analyzed. It is shown that, SF-TEBS4 combined with OCS4 and OCI4, can achieve the asymptotic stability. Moreover, the numerical experiment for determining the first order derivative of a function indicates that (1) the global accuracy of our scheme is fourth-order, and that (2) the computational error is reduced greatly. The numerical experiment for solving the wave propagation problem shows that the combination of SF-TEBS4 with OCS4 and OCI4 can effectively suppress the growth rate of the computational error, preserve the group velocity and the numerical asymptotic stability. The theoretical and numerical analyses show that the combination of SF-TEBS4 with OCS4 and OCI4 can be applied to simulate the propagation of small scale waves.