Boundary-optimized closures for diagonal-norm upwind SBP operators

Abstract

By employing non-equispaced grid points near boundaries, boundary-optimized upwind finite-difference operators of orders up to nine are developed. The boundary closures are constructed within a diagonal-norm summation-by-parts (SBP) framework, ensuring linear stability on piecewise curvilinear multiblock grids. For linear problems, this stability is inherited from the diagonal norm SBP framework provided the approximation satisfies the metric identities (i.e. it is free-stream preserving). For nonlinear problems, the flux-vector splitting should be linear in the metric terms to inherit these stability properties. Boundary and interface conditions are imposed using either weak enforcement through simultaneous approximation terms (SAT) or strong enforcement via the projection method. The proposed operators yield significantly improved accuracy compared with SBP operators constructed on equidistant grids. The resulting SBP--SAT and SBP--projection discretizations produce fully explicit systems of ordinary differential equations. The accuracy and stability properties of the proposed operators are demonstrated through numerical experiments for linear hyperbolic problems in one spatial dimension and for the compressible Euler equations in two spatial dimensions.