The application of finite difference schemes to handle a horizontal interface between two media gives numerically adequate results, as evidenced in the solution of parabolic wave equations with density variations. Using the same finite difference schemes for handling a stair-step approximation to an irregular interface gives satisfactory results provided that the slope angle is small, but a small error is introduced. In the event that the slope angle is not small, this error has to be handled carefully since it may influence the results. Using an analysis of the error, this paper derives a closed form expression for a correction term. An irregular interface can then be handled adequately by the same numerical treatment used for the horizontal interface by applying this correction term. A test case is presented to demonstrate the inaccuracy of using one standard numerical horizontal interface treatment and to show how our new development improves the accuracy.
Energy-conserving finite difference schemes for nonlinear wave equations with dynamic boundary conditions
