A Stable and Conservative Interface Treatment of Arbitrary Spatial Accuracy

Abstract

Stable and accurate interface conditions based on the SAT penalty method are derived for the linear advection–diffusion equation. The conditions are functionally independent of the spatial order of accuracy and rely only on the form of the discrete operator. We focus on high-order finite-difference operators that satisfy the summation-by-parts (SBP) property. We prove that stability is a natural consequence of the SBP operators used in conjunction with the new, penalty type, boundary conditions. In addition, we show that the interface treatments are conservative. The issue of the order of accuracy of the interface boundary conditions is clarified. New finite-difference operators of spatial accuracy up to sixth order are constructed which satisfy the SBP property. These finite-difference operators are shown to admit design accuracy (pth-order global accuracy) when (p−1)th-order stencil closures are used near the boundaries, if the physical boundary conditions and interface conditions are implemented to at leastpth-order accuracy. Stability and accuracy are demonstrated on the nonlinear Burgers' equation for a 12-subdomain problem with randomly distributed interfaces.