Conservative and Stable Degree Preserving SBP Operators for Non-Conforming Meshes

TL;DR

This paper introduces a new class of SBP-SAT operators for non-conforming meshes that preserve the scheme's degree, ensuring high-order accuracy, conservation, and stability without loss of accuracy at interfaces.

Contribution

The authors develop degree preserving SBP-SAT operators that maintain high-order accuracy and stability in non-conforming discretizations, overcoming limitations of existing schemes.

Findings

The new operators preserve the scheme's degree without loss at interfaces.

Mathematical analysis confirms conservation and energy stability.

Numerical tests verify high-order accuracy and stability.

Abstract

Non-conforming numerical approximations offer increased flexibility for applications that require high resolution in a localized area of the computational domain or near complex geometries. Two key properties for non-conforming methods to be applicable to real world applications are conservation and energy stability. The summation-by-parts (SBP) property, which certain finite-difference and discontinuous Galerkin methods have, finds success for the numerical approximation of hyperbolic conservation laws, because the proofs of energy stability and conservation can discretely mimic the continuous analysis of partial differential equations. In addition, SBP methods can be developed with high-order accuracy, which is useful for simulations that contain multiple spatial and temporal scales. However, existing non-conforming SBP schemes result in a reduction of the overall degree of the scheme, which leads to a reduction in the order of the solution error. This loss of degree is due to the particular interface coupling through a simultaneous-approximation-term (SAT). We present in this work a novel class of SBP-SAT operators that maintain conservation, energy stability, and have no loss of the degree of the scheme for non-conforming approximations. The new \emph{degree preserving} discretizations require an ansatz that the norm matrix of the SBP operator is of a degree $\geq 2p$, in contrast to, for example, existing finite difference SBP operators, where the norm matrix is $2p-1$ accurate. We demonstrate the fundamental properties of the new scheme with rigorous mathematical analysis as well as numerical verification.