Chen, J. H., 1973: Numerical boundary conditions and computational modes. Journal of Computational Physics, 13, 522-535.

Abstract: Methods of specifying theboundary conditions for the finite difference approximation of the 1 - D linearized shallow water wave equation are tested. First, extrapolation schemes for boundary values are studied. In particular, "characteristic extrapolations" based on the characteristic equations are proposed for the inflow boundaries. Second, discussions are given on the overall setting of the boundary conditions. An improper treatment of "computational" boundary conditions excites the computational modes in the numerical solutions. Two types of boundary treatments are found to avoid or suppress the unfavorable oscillation due to the computational modes. One type uses order-consistent equation for extrapolations in a well-posed case, and the other uses a local smoothing of the solutions at the boundaries in an over-specified case.