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High order well-balanced and total-energy-conserving local discontinuous Galerkin methods for compressible self-gravitating Euler equations

Liang Pan, Wei Chen, Jianxian Qiu, Tao Xiong

TL;DR

The paper tackles the challenge of simulating compressible self-gravitating Euler flows with a time-dependent gravitational potential by developing a high-order, well-balanced, and total-energy-conserving local DG scheme. Key ideas include decomposing the gravitational potential into equilibrium and perturbation parts to achieve well-balancedness and rewriting the energy equation in a conservative form with a carefully designed gravity flux to ensure total-energy conservation, even in multiple spatial dimensions. The method integrates a multi-dimensional HLLC flux with structure-preserving modifications, a DG-based Poisson solver for gravity, high-order SSP-RK time stepping, and an oscillation-elimination approach to maintain stability near shocks and discontinuities. Comprehensive 2D and 3D numerical tests confirm the scheme’s accuracy, WB property, TEC, shock-capturing strength, and robustness, demonstrating its potential for astrophysical simulations without assuming spherical symmetry.

Abstract

In this paper, we develop a high order structure-preserving local discontinuous Galerkin (DG) scheme for the compressible self-gravitating Euler equations, which pose great challenges due to the presence of time-dependent gravitational potential. The designed scheme is well-balanced for general polytropic equilibrium state and total energy conserving for multiple spatial dimensions without an assumption of spherical symmetry. The well-balanced property is achieved by decomposing the gravitational potential into equilibrium and perturbation parts, employing a modified Harten-Lax-van Leer-contact flux and a modification of the discretization for the source term. Conservation of total energy is particularly challenging in the presence of self-gravity, especially when aiming for high order accuracy. To address this, we rewrite the energy equation into a conservative form, and carefully design an energy flux with the aid of weak formulation from the DG method to maintain conservation as well as high order accuracy. The resulting scheme can be extended to high order in time discretizations. Numerical examples for two and three dimensional problems are provided to verify the desired properties of our proposed scheme, including shock-capturing, high order accuracy, well balance, and total energy conservation.

High order well-balanced and total-energy-conserving local discontinuous Galerkin methods for compressible self-gravitating Euler equations

TL;DR

The paper tackles the challenge of simulating compressible self-gravitating Euler flows with a time-dependent gravitational potential by developing a high-order, well-balanced, and total-energy-conserving local DG scheme. Key ideas include decomposing the gravitational potential into equilibrium and perturbation parts to achieve well-balancedness and rewriting the energy equation in a conservative form with a carefully designed gravity flux to ensure total-energy conservation, even in multiple spatial dimensions. The method integrates a multi-dimensional HLLC flux with structure-preserving modifications, a DG-based Poisson solver for gravity, high-order SSP-RK time stepping, and an oscillation-elimination approach to maintain stability near shocks and discontinuities. Comprehensive 2D and 3D numerical tests confirm the scheme’s accuracy, WB property, TEC, shock-capturing strength, and robustness, demonstrating its potential for astrophysical simulations without assuming spherical symmetry.

Abstract

In this paper, we develop a high order structure-preserving local discontinuous Galerkin (DG) scheme for the compressible self-gravitating Euler equations, which pose great challenges due to the presence of time-dependent gravitational potential. The designed scheme is well-balanced for general polytropic equilibrium state and total energy conserving for multiple spatial dimensions without an assumption of spherical symmetry. The well-balanced property is achieved by decomposing the gravitational potential into equilibrium and perturbation parts, employing a modified Harten-Lax-van Leer-contact flux and a modification of the discretization for the source term. Conservation of total energy is particularly challenging in the presence of self-gravity, especially when aiming for high order accuracy. To address this, we rewrite the energy equation into a conservative form, and carefully design an energy flux with the aid of weak formulation from the DG method to maintain conservation as well as high order accuracy. The resulting scheme can be extended to high order in time discretizations. Numerical examples for two and three dimensional problems are provided to verify the desired properties of our proposed scheme, including shock-capturing, high order accuracy, well balance, and total energy conservation.

Paper Structure

This paper contains 21 sections, 3 theorems, 127 equations, 39 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathmatical Model
  3. Polytropic steady-states and Lane-Emden equation
  4. Total-energy-conserving Property
  5. Numerical schemes
  6. Notations
  7. The multi-dimensional HLLC flux
  8. The standard LDG scheme
  9. The structure-preserving LDG scheme
  10. The modified HLLC numerical flux
  11. Source term approximations
  12. Energy flux $({\mathbf{F}}_g)$ approximations
  13. Flowchart for the structure-preserving LDG scheme
  14. High order time discretizations
  15. Oscillation-eliminating approach
  16. ...and 6 more sections

Key Result

Lemma 3.1

For any two states ${\mathbf{U}}_L = (\rho_L,\mathbf{0},p/(\gamma-1))^T$ and ${\mathbf{U}}_R = (\rho_R,\mathbf{0},p/(\gamma-1))^T$, the multidimensional HLLC flux satisfies:

Figures (39)

  • Figure : (a) Structure-preserving LDG scheme: velocity magnitude $||{\mathbf{u}}||$.
  • Figure : (a) Structure-preserving LDG scheme: velocity magnitude $||{\mathbf{u}}||$.
  • Figure : (a) density $\rho$.
  • Figure : (a) Total energy $E_{tot}$ with time $T$.
  • Figure : (a)
  • ...and 34 more figures

Theorems & Definitions (17)

  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Example 5.1
  • Example 5.2
  • Example 5.3
  • ...and 7 more