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Entropy correction artificial viscosity for high order DG methods using multiple artificial viscosities

Raymond Park, Jesse Chan

Abstract

Entropy stable discontinuous Galerkin (DG) methods display improved robustness for problems with shocks, turbulence, and under-resolved features by enforcing an entropy inequality. Such methods have traditionally relied on entropy conservative (EC) fluxes that are computationally expensive to evaluate. An alternative approach for enforcing an entropy inequality is through a minimally dissipative ``entropy correction" artificial viscosity. We review how to construct such an artificial viscosity formulation and extend this approach to multiple types of viscosity (e.g., viscosity and thermal diffusivity). We determine simple analytical expressions for optimal viscosity parameters. We compare this to the case of a single monolithic viscosity parameter for different 1D and 2D problems, and show that the proposed method allows users to more precisely target specific physical phenomena while retaining robustness for general problem settings.

Entropy correction artificial viscosity for high order DG methods using multiple artificial viscosities

Abstract

Entropy stable discontinuous Galerkin (DG) methods display improved robustness for problems with shocks, turbulence, and under-resolved features by enforcing an entropy inequality. Such methods have traditionally relied on entropy conservative (EC) fluxes that are computationally expensive to evaluate. An alternative approach for enforcing an entropy inequality is through a minimally dissipative ``entropy correction" artificial viscosity. We review how to construct such an artificial viscosity formulation and extend this approach to multiple types of viscosity (e.g., viscosity and thermal diffusivity). We determine simple analytical expressions for optimal viscosity parameters. We compare this to the case of a single monolithic viscosity parameter for different 1D and 2D problems, and show that the proposed method allows users to more precisely target specific physical phenomena while retaining robustness for general problem settings.

Paper Structure

This paper contains 19 sections, 3 theorems, 59 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Nonlinear conservation laws and entropy inequalities
  3. High order discontinuous Galerkin formulations and viscosity discretizations
  4. High order DG formulation
  5. Entropy correction artificial viscosity
  6. DG formulation of monolithic ECAV model
  7. Multiple artificial viscosities approach
  8. DG Formulation of multiple AV models
  9. Optimization problem involving the cell entropy inequality
  10. Viscosity models
  11. Svärd thermal viscosity
  12. Spectral vanishing viscosity (SVV)
  13. Numerical experiments
  14. Compressible Euler equations
  15. Density wave and high order accuracy
  16. ...and 4 more sections

Key Result

lemma 1

Let $\epsilon_k(\boldsymbol{u}_h) \ge 0$, and $\bm{g}_\text{visc}$ be given by DGofv, L2oftheta_sigma, viscous_calc. Then, for a periodic domain, $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Temperature profile of receding flow problem, $T = 0.18$
  • Figure 2: Receding Flow $N = 2, M = 130$
  • Figure 3: Solution to Kelvin Helmholtz instability using SVV and Laplacian viscosity and SVV coefficient at final time $T = 25.0$, and using $N = 3$, $64 \times 64$ grid.
  • Figure 4: $N=3, 64 \times 64, T = 0.14$, log scale. To highlight how small the SVV coefficient is relative to the Laplacian coefficient, we show the heat map of the log scale of the coefficients.
  • Figure : 1D Receding flow initial condition

Theorems & Definitions (10)

  • lemma 1
  • lemma 2
  • remark 1
  • remark 2
  • remark 3
  • lemma 3
  • proof
  • remark 4
  • remark 5
  • remark 6