Nonlinear SBP operators for stable approximations of skew-symmetric hyperbolic initial boundary value problems

Abstract

Summation-by-parts (SBP) operators in combination with simultaneous-approximation-terms (SATs) for boundary conditions provides a recipe that can be applied to well posed linear initial boundary value problems.The SBP-SAT frame work guarantees semi-discrete energy stability and can be applied to most numerical discretization methods. High order accuracy comes without additional difficulty. The extension of SBP-SAT to nonlinear initial boundary value problems has so far been somewhat unclear. However, recent developments have shown how to construct stable nonlinear SAT terms. In this note we develop nonlinear SBP operators. These combined developments provide an SBP-SAT based recipe for stable approximations of nonlinear skew-symmetric hyperbolic initial boundary value problems.