Stable Volume Dissipation for High-Order Finite-Difference and Spectral-Element Methods with the Summation-by-Parts Property (2503.12670v2)

Abstract: The construction of stable, conservative, and accurate volume dissipation is extended to discretizations that possess a generalized summation-by-parts (SBP) property within a tensor-product framework. The dissipation operators can be applied to any finite-difference or spectral-element scheme that uses the SBP framework, including high-order entropy-stable schemes. Additionally, we clarify the incorporation of a variable coefficient within the operator structure and analyze the impact of a boundary correction matrix on operator structure and accuracy. Following the theoretical development and construction of novel dissipation operators, we relate the presented volume dissipation to the use of upwind SBP operators. When applied to spectral-element methods, the presented approach yields unique dissipation operators that can also be derived through alternative approaches involving orthogonal polynomials. Numerical examples featuring the linear convection, Burgers, and Euler equations verify the properties of the constructed dissipation operators and assess their performance compared to existing upwind SBP schemes, including linear stability behaviour. When applied to entropy-stable schemes, the presented approach results in accurate and robust methods that can solve a broader range of problems where comparable existing methods fail.