Accurate Radiation Boundary Conditions for the Linearized Euler Equations in Cartesian Domains

Abstract

We construct, analyze, and test convergent sequences of radiation boundary conditions at planar boundaries for the compressible Euler equations linearized about a uniform, subsonic flow. First we formulate well-posed exact conditions, identifying one which involves only a single nonlocal term at each boundary. Next we develop the basic convergence theory for rational approximations to the symbol of the nonlocal term, which we then specialize to the Pade approximants. Finally, we present numerical experiments using a seventh order implementation of the approximate conditions. The experiments involve the long time propagation of initial pressure pulses in periodic and duct geometry. With these we conclude that [i.] standard low order conditions lead to O(1) errors over moderate times; [ii.] good accuracy can be obtained with sufficiently high order approximate conditions of Pade type; [iii.] the actual errors are consistent with the theory---spectral convergence with increasing order for fixed times but a need to use high order approximations for long time accuracy.