Full-Spectrum Dispersion Relation Preserving Summation-by-Parts Operators

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1565-1588, August 2024.
Abstract. The dispersion error is currently the dominant error for computed solutions of wave propagation problems with high-frequency components. In this paper, we define and give explicit examples of interior [math]-dispersion-relation-preserving schemes, of interior order of accuracy 4, 5, 6, and 7, with a complete methodology to construct them. These are dual-pair finite-difference schemes for systems of hyperbolic partial differential equations which satisfy the summation-by-parts principle and preserve the dispersion relation of the continuous problem uniformly to an [math] error tolerance for their interior stencil. We give a general framework to design provably stable finite-difference operators whose interior stencil preserves the dispersion relation for hyperbolic systems such as the elastic wave equation. The operators we derive here can resolve the highest frequency ([math]-mode) present on any equidistant grid at a tolerance of [math] maximum error within the interior stencil, with minimal extra stencil points. As standard finite-difference schemes have a [math] dispersion error for high-frequency components, fine meshes must be used to resolve these components. Our derived schemes may compute solutions with the same accuracy as traditional schemes on far coarser meshes, which in high dimensions significantly improves the computational cost.