Solutions of Category 1 Problems Using High-Order Difference Methods and Perfectly Matched Layers

We present solutions to the three category 1 benchmark problems: the test of the dispersion characteristics of our difference methods, the scattering of a plane wave of an interface, and the propagation of a pulse through an artificial boundary for a skewed uniform flow and for subsonic Couette flow. Our base numerical method is the combination of maximal order central differences in space with standard fourth order Runge-Kutta methods in time. Near boundaries we add additional mesh points to stabilize the one-sided differencing, which allows us to stably implement methods of orders in the twenties with nonoverlapping multiple domains. For problem 1 we also test Hermite difference methods. These use staggered two-point stencils and time-march via Taylor series with the order increased by carrying derivative data. For the Hermite methods no special treatments of the boundaries are required. For each type of discretization we consider various orders. To truncate the domain we use a multi-parameter family of re.ectionless sponge layers or PMLs. We study the accuracy and stability of the solutions as a function of the layer width and parameters, and the robustness of the parameter choices as the problems change.