New, efficient, and accurate high order derivative and dissipation operators satisfying summation by parts, and applications in three-dimensional multi-block evolutions

TL;DR

This paper develops new high-order finite difference operators satisfying summation by parts, optimized for minimal bandwidth, boundary error, or spectral radius, and demonstrates their stability and accuracy in 3D wave simulations.

Contribution

It introduces a set of optimized, high-order difference and dissipation operators satisfying summation by parts, with applications in multi-block 3D evolutions.

Findings

Operators achieve up to tenth order accuracy in the interior.

Optimization improves spectral radius and accuracy significantly.

Stable and accurate 3D wave evolution demonstrated.

Abstract

We construct new, efficient, and accurate high-order finite differencing operators which satisfy summation by parts. Since these operators are not uniquely defined, we consider several optimization criteria: minimizing the bandwidth, the truncation error on the boundary points, the spectral radius, or a combination of these. We examine in detail a set of operators that are up to tenth order accurate in the interior, and we surprisingly find that a combination of these optimizations can improve the operators' spectral radius and accuracy by orders of magnitude in certain cases. We also construct high-order dissipation operators that are compatible with these new finite difference operators and which are semi-definite with respect to the appropriate summation by parts scalar product. We test the stability and accuracy of these new difference and dissipation operators by evolving a three-dimensional scalar wave equation on a spherical domain consisting of seven blocks, each discretized with a structured grid, and connected through penalty boundary conditions.