Optimal boundary closures for diagonal-norm upwind SBP operators

TL;DR

This paper introduces boundary-optimized upwind finite-difference operators within a diagonal-norm SBP framework, improving accuracy and efficiency for hyperbolic PDEs on complex grids.

Contribution

It develops high-order boundary closures on non-equispaced grids within a diagonal-norm SBP framework, enhancing stability and accuracy for hyperbolic problems.

Findings

Significantly improved accuracy over equidistant grid operators

Enhanced computational efficiency in boundary treatments

Stable explicit discretizations for hyperbolic PDEs

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. 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 and computational efficiency 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.