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Volume Term Adaptivity for Discontinuous Galerkin Schemes

Daniel Doehring, Jesse Chan, Hendrik Ranocha, Michael Schlottke-Lakemper, Manuel Torrilhon, Gregor Gassner

Abstract

We introduce the concept of volume term adaptivity for high-order discontinuous Galerkin (DG) schemes solving time-dependent partial differential equations. Termed v-adaptivity, we present a novel general approach that exchanges the discretization of the volume contribution of the DG scheme at every Runge-Kutta stage based on suitable indicators. Depending on whether robustness or efficiency is the main concern, different adaptation strategies can be chosen. Precisely, the weak form volume term discretization is used instead of the entropy-conserving flux-differencing volume integral whenever the former produces more entropy than the latter, resulting in an entropy-stable scheme. Conversely, if increasing the efficiency is the main objective, the weak form volume integral may be employed as long as it does not increase entropy beyond a certain threshold or cause instabilities. Thus, depending on the choice of the indicator, the v-adaptive DG scheme improves robustness, efficiency and approximation quality compared to schemes with a uniform volume term discretization. We thoroughly verify the accuracy, linear stability, and entropy-admissibility of the v-adaptive DG scheme before applying it to various compressible flow problems in two and three dimensions.

Volume Term Adaptivity for Discontinuous Galerkin Schemes

Abstract

We introduce the concept of volume term adaptivity for high-order discontinuous Galerkin (DG) schemes solving time-dependent partial differential equations. Termed v-adaptivity, we present a novel general approach that exchanges the discretization of the volume contribution of the DG scheme at every Runge-Kutta stage based on suitable indicators. Depending on whether robustness or efficiency is the main concern, different adaptation strategies can be chosen. Precisely, the weak form volume term discretization is used instead of the entropy-conserving flux-differencing volume integral whenever the former produces more entropy than the latter, resulting in an entropy-stable scheme. Conversely, if increasing the efficiency is the main objective, the weak form volume integral may be employed as long as it does not increase entropy beyond a certain threshold or cause instabilities. Thus, depending on the choice of the indicator, the v-adaptive DG scheme improves robustness, efficiency and approximation quality compared to schemes with a uniform volume term discretization. We thoroughly verify the accuracy, linear stability, and entropy-admissibility of the v-adaptive DG scheme before applying it to various compressible flow problems in two and three dimensions.
Paper Structure (30 sections, 63 equations, 25 figures, 1 table)

This paper contains 30 sections, 63 equations, 25 figures, 1 table.

Table of Contents

  1. Introduction
  2. Volume Term Formulations in Nodal Discontinuous Galerkin Schemes
  3. Weak Formulation of the 1D Discontinuous Galerkin Spectral Element Method
  4. Flux--Differencing Formulation of the 1D Discontinuous Galerkin Spectral Element Method
  5. Indicator
  6. Rigorous Indicator for more Robust Entropy--Stable Schemes
  7. Heuristic Indicator for more Efficient Schemes
  8. Shock--Capturing Indicator
  9. Shock--Capturing via Blended DG--FV
  10. Combining Shock--Capturing and Entropy--Production Indicators
  11. Verification
  12. Convergence
  13. Isentropic Vortex Advection with Viscous Surface Flux
  14. Density Wave with Inviscid Surface Flux
  15. Linear Stability
  16. ...and 15 more sections

Figures (25)

  • Figure 1: Errors of the isentropic vortex advection testcase with viscous surface flux.
  • Figure 2: Errors for the density wave convergence testcase with inviscid surface flux.
  • Figure 3: Spectra at initial time $t=0$ for the linear stability testcase.
  • Figure 4: Entropy difference over time for the density wave linear stability testcase with rigorous indicator for the adaptive volume term discretization. The simulations are performed with CFL $=0.9$.
  • Figure 5: Shock formation and propagation for Burgers' equation \ref{['eq:BurgersEquation']} at $t_f = 0.25$. Note the different scalings of the vertical axis.
  • ...and 20 more figures