# Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization

## TL;DR

This paper extends entropy correction techniques for residual distribution and discontinuous Galerkin schemes, integrating them into the SBP-SAT framework, and develops fully discrete entropy-preserving schemes with explicit solutions and numerical validation.

## Contribution

It generalizes entropy correction methods within the SBP-SAT framework, introduces explicit solutions for optimization problems, and applies these to develop fully discrete entropy-preserving residual distribution schemes.

## Key findings

- Explicit solutions for correction terms are derived.
- The approach is successfully applied to Euler equations.
- Numerical experiments confirm the effectiveness of the schemes.

## Abstract

For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/ dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall (J.~Comp.~Phys. 372: pp. 640--666, 2018). In this work, the correction terms are characterized as solutions of certain optimization problems and are adapted to the SBP-SAT framework, focusing on discontinuous Galerkin methods. Novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. For all of these optimization problems, explicit solutions are provided. Additionally, the correction approach is applied for the first time to obtain a fully discrete entropy conservative/dissipative RD scheme. Here, the application of the deferred correction (DeC) method for the time integration is essential. This paper can be seen as describing a systematic method to construct structure preserving discretization, at least for the considered example.