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Invariant-domain preserving IMEX schemes for the nonequilibrium Gray Radiation-Hydrodynamics equations Part I

Jean-Luc Guermond, Eric J. Tovar

TL;DR

The paper develops a first-order IMEX scheme for nonequilibrium gray radiation-hydrodynamics that preserves an invariant domain and conserves energy. It achieves this by decomposing the GRH system into two hyperbolic subsystems and a parabolic subsystem, with an additive splitting that guarantees IDP and conservation for all iterations. A robust fixed-point approach is used to handle the parabolic stage, and a detailed Riemann-problem analysis yields guaranteed wave-speed bounds to ensure IDP. Numerical tests on Marshak waves, 1D radiative shocks, and ICF-like configurations demonstrate correct first-order convergence and robust behavior across regimes. Overall, the work provides a solid, provably IDP foundation for gravity-free GRH simulations and paves the way for higher-order extensions in a subsequent paper.

Abstract

In this work we introduce an implicit-explicit invariant-domain preserving approximation of the nonequilibrium gray radiation-hydrodynamics equations. A time and space approximation of the system is proposed using a novel split of the equations composed of three elementary subsystems, two hyperbolic and one parabolic. The approximation thus realized is proved to be consistent, conservative, invariant-domain preserving, and first-order accurate. The proposed method is a stepping stone for achieving higher-order accuracy in space and time in the forthcoming second part of this work. The method is numerically illustrated and shown to converge as advertised. This paper is dedicated to the memory of Peter Lax.

Invariant-domain preserving IMEX schemes for the nonequilibrium Gray Radiation-Hydrodynamics equations Part I

TL;DR

The paper develops a first-order IMEX scheme for nonequilibrium gray radiation-hydrodynamics that preserves an invariant domain and conserves energy. It achieves this by decomposing the GRH system into two hyperbolic subsystems and a parabolic subsystem, with an additive splitting that guarantees IDP and conservation for all iterations. A robust fixed-point approach is used to handle the parabolic stage, and a detailed Riemann-problem analysis yields guaranteed wave-speed bounds to ensure IDP. Numerical tests on Marshak waves, 1D radiative shocks, and ICF-like configurations demonstrate correct first-order convergence and robust behavior across regimes. Overall, the work provides a solid, provably IDP foundation for gravity-free GRH simulations and paves the way for higher-order extensions in a subsequent paper.

Abstract

In this work we introduce an implicit-explicit invariant-domain preserving approximation of the nonequilibrium gray radiation-hydrodynamics equations. A time and space approximation of the system is proposed using a novel split of the equations composed of three elementary subsystems, two hyperbolic and one parabolic. The approximation thus realized is proved to be consistent, conservative, invariant-domain preserving, and first-order accurate. The proposed method is a stepping stone for achieving higher-order accuracy in space and time in the forthcoming second part of this work. The method is numerically illustrated and shown to converge as advertised. This paper is dedicated to the memory of Peter Lax.
Paper Structure (42 sections, 11 theorems, 84 equations, 8 figures, 4 tables, 1 algorithm)

This paper contains 42 sections, 11 theorems, 84 equations, 8 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The model
  3. Governing equations
  4. Thermodynamics and invariant domain
  5. Three elementary subsystems
  6. IMEX split (hyperbolic-parabolic)
  7. Hyperbolic stage 1
  8. Hyperbolic stage 2
  9. Parabolic stage
  10. Approximation details
  11. First-order IDP scheme
  12. Hyperbolic stage 1
  13. Hyperbolic stage 2
  14. Multiplicative vs. additive splitting
  15. Multiplicative splitting
  16. ...and 27 more sections

Key Result

Lemma 5.1

Assume that the space discretization meets the structural assumptions ass1--ass5 from Assumption Ass:assumption_on_space_discretization. Assume that $\textup{u}_i^n$ is in ${\mathcal{A}}(b)$ for all $i\in{\mathcal{V}}$. Assume that the time step satisfies ${\tau}\le \max_{i\in{\mathcal{V}}}$. Let $\

Figures (8)

  • Figure 1: Marshak wave with $\sigma_t=300(\frac{T_{\textup{ref}}}{T})^3$ at $T={\qty[scientific-notation=false, round-mode=figures,round-precision = 5, drop-zero-decimal, round-pad = false]{0.02}{sh}}$. Left: temperature. Center: zoom in on temperature shock location. Right: radiation energy.
  • Figure 2: Mach 3 radiative shock with $\sigma_t = {\qty[scientific-notation=false, round-mode=figures,round-precision = 5, drop-zero-decimal, round-pad = false]{500}{{cm}^{-1}}}$ at $T={\qty[scientific-notation=false, round-mode=figures,round-precision = 5, drop-zero-decimal, round-pad = false]{1}{sh}}$. Left: velocity. Center: temperature. Right: radiation energy.
  • Figure 3: Mach 3 radiative shock with $\sigma_a=500 \frac{\rho}{\rho_\text{ref}}\left(\frac{T_\text{ref}}{T}\right)^{3.5}{\qty[scientific-notation=false, round-mode=figures,round-precision = 5, drop-zero-decimal, round-pad = false]{}{cm^{-1}}}$ and $\sigma_s = 0$ for Mach 3. Left: velocity. Center: temperature. Right: radiation energy.
  • Figure 4: Mach 50 radiative shock with $\sigma_a=\sigma_t = {\qty[scientific-notation=false, round-mode=figures,round-precision = 5, drop-zero-decimal, round-pad = false]{500}{{cm}^{-1}}}$ at $t={\qty[scientific-notation=false, round-mode=figures,round-precision = 5, drop-zero-decimal, round-pad = false]{10}{sh}}$. Left: velocity. Center: temperature. Right: radiation energy.
  • Figure 5: 1D ICF. Left: time history of maximum density. Center: density profile at the time maximum density is reached. Right: temperature at the time maximum density is reached. Mesh3 has $4097$ grid points and Mesh8 has $131073$ grid points.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Remark 2.1: Total energy
  • Remark 2.2: Internal energy
  • Remark 2.3: Nonconservative products
  • Remark 2.4: Thermodynamic nonequilibrium
  • Definition 4.2: Conservation
  • Lemma 5.1: ${\boldsymbol u}^n\mapsto \textup{u}^{n,1}$ is IDP & conservative
  • Proof 1
  • Remark 5.2: Density update
  • Lemma 5.3: ${\boldsymbol u}^{n}\mapsto \textup{u}^{n,2}$ is IDP & conservative
  • Proof 2
  • ...and 16 more