Optimal diagonal-norm SBP operators

Optimal boundary closures are derived for first derivative, finite difference operators of order 2, 4, 6 and 8. The closures are based on a diagonal-norm summation-by-parts (SBP) framework, thereby guaranteeing linear stability on piecewise curvilinear multi-block grids and entropy stability for nonlinear equations that support a convex extension. The new closures are developed by enriching conventional approaches with additional boundary closure stencils and non-equidistant grid distributions at the domain boundaries. Greatly improved accuracy is achieved near the boundaries, as compared with traditional diagonal-norm operators of the same order. The superior accuracy of the new optimal diagonal-norm SBP operators is demonstrated for linear hyperbolic systems in one dimension and for the nonlinear compressible Euler equations in two dimensions.