A class of energy stable, high-order finite-difference interface schemes suitable for adaptive mesh refinement of hyperbolic problems

Abstract

We present a class of energy stable, high-order finite-difference interface closures for grids with step resolution changes. These grids are commonly used in adaptive mesh refinement of hyperbolic problems. The interface closures are such that the global accuracy of the numerical method is that of the interior stencil. The summation-by-parts property is built into the stencil construction and implies asymptotic stability by the energy method while being non-dissipative. We present one-dimensional closures for fourth-order explicit and compact Padé type, finite differences. Tests on the scalar one- and two-dimensional wave equations, the one-dimensional Navier–Stokes solution of a shock and two-dimensional inviscid compressible vortex verify the accuracy and stability of this class of methods.