Stable Coupling of Nonconforming, High-Order Finite Difference Methods

TL;DR

This paper introduces a stable method for coupling high-order finite difference methods across nonconforming interfaces, ensuring energy stability and compatibility with different discretization techniques.

Contribution

It develops projection operators for nonconforming coupling that are compatible with energy norms, enabling stable integration of finite difference and discontinuous Galerkin methods.

Findings

Proven energy stability for coupled methods on curved interfaces

Constructed projection operators compatible with summation-by-parts norms

Numerical simulations confirm theoretical stability and eigenvalue analysis

Abstract

A methodology for handling block-to-block coupling of nonconforming, multiblock summation-by-parts finite difference methods is proposed. The coupling is based on the construction of projection operators that move a finite difference grid solution along an interface to a space of piecewise defined functions; we specifically consider discontinuous, piecewise polynomial functions. The constructed projection operators are compatible with the underlying summation-by-parts energy norm. Using the linear wave equation in two dimensions as a model problem, energy stability of the coupled numerical method is proven for the case of curved, nonconforming block-to-block interfaces. To further demonstrate the power of the coupling procedure, we show how it allows for the development of a provably energy stable coupling between curvilinear finite difference methods and a curved-triangle discontinuous Galerkin method. The theoretical results are verified through numerical simulations on curved meshes as well as eigenvalue analysis.