Experiments with stable, high-order difference approximations to hyperbolic initial-boundary value problems

We investigate a simple but seemingly effective approach to deriving stable boundary closures for high-order difference approximations to hyperbolic systems. Precisely, we modify the uniform mesh in a small, 2 - 7 point, layer near the boundaries. Choosing the modified nodes to correspond to Alpert's Gauss-Trapezoidal quadrature nodes, stable schemes of all orders up to 12 are derived. A spectral analysis of the resulting semidiscrete systems is given, along with numerical experiments in two space dimensions.