Targeted Entropy-Stable Nonlinear Artificial Dissipation for High-Order Spectral-Element Summation-By-Parts Discretizations
Abstract
Artificial dissipation improves the robustness of high-order methods for under-resolved flows, but can compromise accuracy and suppress physically relevant structures if its strength and spatial distribution are not appropriately controlled. We present a sensor-activated nonlinear artificial-dissipation framework, inspired by the residual correction method, for continuous and discontinuous summation-by-parts discretizations. In contrast to many existing spectral-element dissipation operators that either reduce the formal order of accuracy or target only the highest polynomial modes, the proposed framework permits design- and higher-order dissipation with explicit control over its magnitude, spatial distribution, activation, and convergence rate under mesh refinement. This modular construction also recovers several existing methods through appropriate choices of its components. The dissipation preserves discrete conservation and semidiscrete entropy stability whenever the base method satisfies these properties. It is dimensionally consistent, element local, and discretization agnostic, as the added term is constructed independently of the base spatial residual. For smooth solutions under suitable mesh assumptions, we establish parameter choices that retain the formal accuracy of the base discretization. Numerical experiments with summation-by-parts discretizations of the Burgers and Euler equations demonstrate substantially improved robustness for under-resolved flows and flows with shock waves while retaining high-order accuracy for smooth solutions, using the same parameter choices across several test problems. Overall, these results show that targeted dissipation can improve the robustness of high-order methods without excessive damping of resolved flow structures.
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BibTeX
@misc{worku2026targeted,
title = {{Targeted Entropy-Stable Nonlinear Artificial Dissipation for High-Order Spectral-Element Summation-By-Parts Discretizations}},
author = {Zelalem Arega Worku and Alex Bercik and David C. Del Rey Fernández and David W. Zingg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17567},
eprint = {2610.17567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17567},
}
RIS
TY - UNPB
TI - Targeted Entropy-Stable Nonlinear Artificial Dissipation for High-Order Spectral-Element Summation-By-Parts Discretizations
AU - Zelalem Arega Worku
AU - Alex Bercik
AU - David C. Del Rey Fernández
AU - David W. Zingg
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17567
UR - https://arxiv.org/abs/2610.17567
AB - Artificial dissipation improves the robustness of high-order methods for under-resolved flows, but can compromise accuracy and suppress physically relevant structures if its strength and spatial distribution are not appropriately controlled. We present a sensor-activated nonlinear artificial-dissipation framework, inspired by the residual correction method, for continuous and discontinuous summation-by-parts discretizations. In contrast to many existing spectral-element dissipation operators that either reduce the formal order of accuracy or target only the highest polynomial modes, the proposed framework permits design- and higher-order dissipation with explicit control over its magnitude, spatial distribution, activation, and convergence rate under mesh refinement. This modular construction also recovers several existing methods through appropriate choices of its components. The dissipation preserves discrete conservation and semidiscrete entropy stability whenever the base method satisfies these properties. It is dimensionally consistent, element local, and discretization agnostic, as the added term is constructed independently of the base spatial residual. For smooth solutions under suitable mesh assumptions, we establish parameter choices that retain the formal accuracy of the base discretization. Numerical experiments with summation-by-parts discretizations of the Burgers and Euler equations demonstrate substantially improved robustness for under-resolved flows and flows with shock waves while retaining high-order accuracy for smooth solutions, using the same parameter choices across several test problems. Overall, these results show that targeted dissipation can improve the robustness of high-order methods without excessive damping of resolved flow structures.
ER -