Simultaneous Approximation Terms for Multi-dimensional Summation-by-Parts Operators

This paper continues our effort to generalize summation-by-parts (SBP) finite-difference methods beyond tensor-products in multiple dimensions. In this work, we focus on the accurate and stable coupling of elements in the context of discontinuous solution spaces. We show how penalty terms — simultaneous approximation terms (SATs) — can be adapted to discretizations based on multi-dimensional SBP operators. We show that the part of the SBP operator corresponding to boundary integration can be decomposed into an interpolation/extrapolation operator and a boundary cubature. The SBP operators themselves are independent of the boundary cubature, and no additional degrees of freedom are introduced. The resulting decomposition facilitates the construction of SATs between arbitrary elements, and we prove that the resulting SBP-SAT discretizations are conservative and stable for divergence-free linear advection. The SATs are illustrated using triangular-element SBP operators with and without nodes that lie on the boundary. The solution accuracy of the resulting SBP-SAT discretizations is verified, and functional accuracy is shown to be superconvergent. The conservation and stability properties of the discretizations are confirmed on a divergence-free linear advection problem.