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arXiv · 2507.14488

Entropy Stable Nodal Discontinuous Galerkin Methods via Quadratic Knapsack Limiting

Brian Christner·Jesse Chan
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Abstract

Lin, Chan (High order entropy stable discontinuous Galerkin spectral element methods through subcell limiting, 2024) enforces a cell entropy inequality for nodal discontinuous Galerkin methods by combining flux corrected transport (FCT)-type limiting and a knapsack solver, which determines optimal limiting coefficients that result in a semi-discrete cell entropy inequality while preserving nodal bounds. In this work, we provide a slight modification of this approach, where we utilize a quadratic knapsack problem instead of a standard linear knapsack problem. We prove that this quadratic knapsack problem can be reduced to efficient scalar root-finding. Numerical results demonstrate that the proposed quadratic knapsack limiting strategy is efficient and results in a semi-discretization with improved regularity in time compared with linear knapsack limiting, while resulting in fewer adaptive timesteps in shock-type problems.

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Brian Christner, Jesse Chan. 2025-07-19. Entropy Stable Nodal Discontinuous Galerkin Methods via Quadratic Knapsack Limiting. https://arxiv.org/abs/2507.14488

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