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

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.

💡 Research Summary

This paper introduces a novel first‑order implicit‑explicit (IMEX) scheme for the nonequilibrium gray radiation‑hydrodynamics (GRH) equations that rigorously preserves the invariant domain (ID) of physically admissible states. The GRH system couples the compressible Euler equations with a diffusion‑type equation for the radiation energy density, featuring stiff source terms that model absorption, emission, and radiation pressure. Direct explicit time integration would require prohibitively small time steps because of the light‑speed term, while a fully implicit treatment would be computationally expensive due to the strong nonlinearity of the opacity functions.

To overcome these difficulties, the authors propose a three‑stage operator split. The first hyperbolic stage isolates the mechanical pressure contribution, yielding a standard Euler system augmented by a linear conservation law for the radiation energy. The second hyperbolic stage handles the radiation pressure terms; by exploiting the fact that radiation pressure does not affect the material internal energy, the authors rewrite the subsystem in a conservative form where both the internal energy and a combined kinetic‑radiation energy remain constant. The third stage is parabolic, containing the stiff absorption‑emission source and the diffusion term. It is discretized with backward Euler in time, and a fixed‑point Picard iteration combined with a local Newton solve updates the radiation energy and material temperature simultaneously.

A key contribution is the rigorous proof that each stage, and consequently the full IMEX algorithm, is consistent, conservative, first‑order accurate, and invariant‑domain preserving. The invariant domain A(b) is defined as the set of states with positive density, bounded compressibility (1 − b ρ > 0), internal energy above a cold‑curve bound, and positive radiation energy. Under mild structural assumptions on the equation of state (the “oracle”)—including quasiconcavity of the cold curve and positivity of heat capacity—the authors show that the numerical solution never leaves A(b). Lemmas 5.1 and 5.3 establish IDP for the two hyperbolic stages, Lemma 5.5 does the same for the parabolic stage, and Theorem 5.6 combines them into a global result.

Spatial discretization uses standard finite‑volume or discontinuous‑Galerkin techniques for the hyperbolic subsystems, with a Riemann solver that respects the derived maximum wave speed for the radiation pressure stage. The diffusion term is treated with a central difference operator, preserving the parabolic regularization’s invariant‑domain property.

Numerical experiments confirm the theoretical findings. Convergence tests demonstrate first‑order accuracy in both L¹ and L² norms. A high‑Mach number test shows that the fixed‑point iteration remains robust even when the flow is strongly compressible, and the total energy is conserved up to machine precision under closed‑boundary conditions.

Overall, the paper provides the first systematic IDP‑IMEX framework for the full nonequilibrium gray radiation‑hydrodynamics system. It lays a solid foundation for the forthcoming second part, where the authors plan to develop high‑order spatial reconstructions and limiting strategies that retain the invariant‑domain property. The methodology is also promising for extensions to multigroup radiation, non‑gray opacities, and more complex equations of state, making it a valuable contribution to computational radiation‑hydrodynamics.