Abstract
We analyze in this paper spurious reflection phenomena that occur at the interface between coarse and fine mesh in the numerical approximation of hyperbolic equations with finite difference method. The principal tool used is a description of propagation properties of numerical solutions by means of their time - Fourier transforms - The corresponding mathematics are given in the companion paper with which the reader should be familiar. In section 4, an exact analytic expression of reflection at the interface for the "standard" finite difference treatment is derived. It is given as an integral in the frequency domain, where a reflection ratio function is defined. In section 3, it is shown that the standard finite difference treatment of the interface point results in conversation of the energy of the numerical solution - Thus (section 7), the analytic expression of reflection is shown to be also that which ensures continuity of energy flows through the mesh interface. In section 8, we show that the global convergence rate is (0(h2) in spite of the fact that the truncation error at the interface point is only 0 (h). In section 9, we examine in the same vein a modified treatment of the interface that was first proposed in a paper by Browning, Kreiss and Oliger. While the amplitude of reflected solutions for this modified scheme converge to zero as 0(k3), the global accuracy of the calculation is 0(h2)as in the standard case.