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On the choice of viscous discontinuous Galerkin discretization for entropy correction artificial viscosity methods

Samuel Q. Van Fleet, Jesse Chan

Abstract

Entropy correction artificial viscosity (ECAV) is an approach for enforcing a semi-discrete entropy inequality through an entropy dissipative correction term. The resulting method can be implemented as an artificial viscosity with an extremely small viscosity coefficient. In this work, we analyze ECAV when the artificial viscosity is discretized using a local discontinuous Galerkin (LDG) method. We prove an $O(h)$ upper bound on the ECAV coefficient, indicating that ECAV does not result in a restrictive time-step condition. We additionally show that ECAV is contact preserving, and compare ECAV to traditional shock capturing artificial viscosity methods.

On the choice of viscous discontinuous Galerkin discretization for entropy correction artificial viscosity methods

Abstract

Entropy correction artificial viscosity (ECAV) is an approach for enforcing a semi-discrete entropy inequality through an entropy dissipative correction term. The resulting method can be implemented as an artificial viscosity with an extremely small viscosity coefficient. In this work, we analyze ECAV when the artificial viscosity is discretized using a local discontinuous Galerkin (LDG) method. We prove an upper bound on the ECAV coefficient, indicating that ECAV does not result in a restrictive time-step condition. We additionally show that ECAV is contact preserving, and compare ECAV to traditional shock capturing artificial viscosity methods.
Paper Structure (19 sections, 4 theorems, 80 equations, 8 figures, 2 tables)

This paper contains 19 sections, 4 theorems, 80 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Nonlinear conservation laws and entropy inequalities
  3. Local discontinuous Galerkin (LDG) discretization of artificial viscosity
  4. High order DG formulation
  5. Semi-discrete entropy estimate
  6. An entropy dissipative viscous LDG formulation
  7. Properties of ECAV under LDG
  8. Estimates for the LDG gradient
  9. Upper bounds on the ECAV artificial viscosity coefficient
  10. Numerical experiments
  11. 2D Inviscid Burgers Equation
  12. Compressible Euler Equations
  13. Shu Isentropic Euler Vortex Problem
  14. Stationary Contact Wave and Contact Preservation
  15. Shock-vortex interaction
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\bm{g}_{\text{visc}}$ be given by eq: LDG 1, eq: LDG 2, and eq: LDG 3. Then, for a periodic domain, where $\Pi_N\bm{v}(\bm{u}_h)$ denotes the $L^2$ projection of the entropy variables.

Figures (8)

  • Figure 1: Example \ref{['Example: Burgers Equation']}, time evolution of the $\max_k{\epsilon_k}$ (semi-log plot) and change in entropy over time for BR-1 and LDG discretizations of the entropy correction artificial viscosity with $N = 2$ and $K = 2 \times 40^2$ elements.
  • Figure 2: Example \ref{['ex: stationary contact wave']}, semi-log plots of the $L^2$ error evolution for two different stationary contact solutions.
  • Figure 3: Example \ref{['ex: Shu Osher']}, density profiles and numerical Schlieren plots.
  • Figure 4: Example \ref{['ex: 1D Density Wave']}, time evolution of the $\max_k{\epsilon_k}$ (semi-log plot) using shock capturing (labeled "SC") and entropy correction artificial viscosity (labeled "ECAV") for degree $N=5$ and both $K=8$ and $K = 16$ elements.
  • Figure 5: Example \ref{['ex: 1D Density Wave']}, time evolution of the $L^2$ error using shock capturing (labeled "SC") and entropy correction artificial viscosity (labeled "ECAV") for degree $N=5$ and both $K=8$ and $K = 16$ elements.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2: Lemma 2 in chan2025artificial
  • Lemma 3
  • proof
  • Remark 1
  • Lemma 4
  • proof
  • Remark 2: On quadrature accuracy
  • Remark 3