Entropy Stable Element-Local Continuous-Discontinuous Residual Correction Summation-By-Parts Discretization

Abstract

Residual correction provides an efficient alternative to two-point volume flux differencing for enforcing entropy stability in high-order discretizations of hyperbolic partial differential equations. We develop an element-local residual-correction summation-by-parts (SBP) method that combines continuous-SBP (C-SBP) and discontinuous-SBP (D-SBP) coupling on conforming meshes. The construction retains continuous assembly without simultaneous approximation terms (SATs) wherever suitable, restricts SATs to selected discontinuous facets, and permits the coupling to change in time as a foundation for adaptive discretizations. The scalar entropy defect and a graph-based distribution vector acting on conservative variables are computed entirely from element-local data, without neighboring states, auxiliary gradients, or correction SATs. The same correction applies to every facet partition, including the pure C-SBP and D-SBP limits. For fixed partitions, we establish conservation and entropy identities for each element residual, exact global conservation, entropy stability up to an explicit regularization defect, and design-order consistency. Changes in the partition are handled by a mass-weighted orthogonal projection that preserves conservative moments and convex admissible sets, does not increase entropy, and does not amplify error in the broken norm. Numerical experiments in one to three dimensions show that the hybrid method retains the smooth-solution accuracy of its D-SBP endpoint under repeated repartitioning. The nonlinear tests demonstrate robustness comparable to Hadamard-form two-point-flux discretizations, particularly when coupled with artificial dissipation.

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BibTeX RIS

BibTeX

@misc{worku2026entropy,
  title = {{Entropy Stable Element-Local Continuous-Discontinuous Residual Correction Summation-By-Parts Discretization}},
  author = {Zelalem Arega Worku and David C. Del Rey Fernandez},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25388},
  eprint = {2610.25388},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25388},
}

RIS

TY  - UNPB
TI  - Entropy Stable Element-Local Continuous-Discontinuous Residual Correction Summation-By-Parts Discretization
AU  - Zelalem Arega Worku
AU  - David C. Del Rey Fernandez
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25388
UR  - https://arxiv.org/abs/2610.25388
AB  - Residual correction provides an efficient alternative to two-point volume flux differencing for enforcing entropy stability in high-order discretizations of hyperbolic partial differential equations. We develop an element-local residual-correction summation-by-parts (SBP) method that combines continuous-SBP (C-SBP) and discontinuous-SBP (D-SBP) coupling on conforming meshes. The construction retains continuous assembly without simultaneous approximation terms (SATs) wherever suitable, restricts SATs to selected discontinuous facets, and permits the coupling to change in time as a foundation for adaptive discretizations. The scalar entropy defect and a graph-based distribution vector acting on conservative variables are computed entirely from element-local data, without neighboring states, auxiliary gradients, or correction SATs. The same correction applies to every facet partition, including the pure C-SBP and D-SBP limits. For fixed partitions, we establish conservation and entropy identities for each element residual, exact global conservation, entropy stability up to an explicit regularization defect, and design-order consistency. Changes in the partition are handled by a mass-weighted orthogonal projection that preserves conservative moments and convex admissible sets, does not increase entropy, and does not amplify error in the broken norm. Numerical experiments in one to three dimensions show that the hybrid method retains the smooth-solution accuracy of its D-SBP endpoint under repeated repartitioning. The nonlinear tests demonstrate robustness comparable to Hadamard-form two-point-flux discretizations, particularly when coupled with artificial dissipation.
ER  -

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