Non-reflecting Boundary Conditions for Wave Propagation Problems

Abstract: We consider two aspects of non-reflecting boundaryconditions for wave propagation problems. First we evaluate aproposed Perfectly Matched Layer (PML) method for thesimulation of advective acoustics. It is shown that theproposed PML becomes unstable for a certain combination ofparameters. A stabilizing procedure is proposed andimplemented. By numerical experiments the performance of thePML for a problem with nonuniform flow is investigated. Furtherthe performance for different types of waves, vorticity andsound waves, are investigated.The second aspect concerns spurious waves, which areintroduced by any discretization procedure. We constructdiscrete boundary conditions, that are nonreflecting for bothphysical and spurious waves, when combined with a fourth orderaccurate explicit discretization of one-way wave equations. Theboundary condition is shown to be GKS-stable. The boundaryconditions are extended to hyperbolic systems in two spacedimensions, by combining exact continuous non-reflectingboundary conditions and the one dimensional discretelynon-reflecting boundary condition. The resulting boundarycondition is localized by the standard Pad´eapproximation.Numerical experiments reveal that the resulting methodsuffers from boundary instabilities. Analysis of a relatedcontinuous problem suggests that the discrete boundarycondition can be stabilized by adding tangential viscosity atthe boundary. For the lowest order Pad´e approximation weare able to stabilize the discrete boundary condition.