To the boundary and back—a numerical study

Abstract

This study identifies the key parameters upon which energy absorption at artificial boundaries depends. A thorough numerical study is presented, of typical reflections from open computational boundaries, for problems governed by hyperbolic systems of equations. The emphasis is on systems, where it is often the combination of all boundary procedures that determine the quality of boundary treatment. We study dissipative numerical models which have so far not been analysed to the same extent as non-dissipative models and employ a Law-Wendroff-type scheme as a prototype. While it is widely accepted that dissipative models tend to give fewer problems than non-dissipative ones, we show a variety of cases where substantial reflections do occur even in ID and quasi-ID set-ups, where theory predicts best results. This can partly be explained by the vanishing of dissipation in the far field. Group velocity analysis, justifiable on the grounds of weak dissipation, predicts a pathological behaviour which is confirmed by numerical experiments. We demonstrate strong focusing of asymptotic errors generated at the artificial boundary. Internal reflections due to slowly expanding grids are shown for non-linear systems. The need for high-frequency boundary conditions naturally arises and combined low-high-frequency boundary recipes following Higdon, Vichnevetsky and Pariser are adapted to systems and tested. Partial cures are also discussed, mainly in terms of pointing out their theoretically limited potential.