Entropy Stable Nodal Discontinuous Galerkin Methods via Quadratic Knapsack Limiting

TL;DR

This paper introduces a quadratic knapsack limiting approach for entropy stable nodal discontinuous Galerkin methods, improving regularity and efficiency in shock problems by reducing to scalar root-finding.

Contribution

It proposes a novel quadratic knapsack problem formulation for limiting, enhancing stability and efficiency over previous linear methods in high-order DG schemes.

Findings

Improved regularity in time discretization.

Fewer adaptive timesteps needed in shock problems.

Efficient scalar root-finding implementation.

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.