Implicit-explicit Runge-Kutta for radiation hydrodynamics I: gray diffusion

TL;DR

This paper introduces a LIMEX-RK-based time integration framework for non-relativistic radiation hydrodynamics with gray diffusion, enabling second-order (and higher) accuracy without nonlinear iterations between hydrodynamics and radiation. By duplicating the state to form LIMEX partitions and deriving a temperature closure, the authors achieve stable, energy-conserving integration with explicit hydrodynamics and linearly implicit radiation coupling. They prove local and global energy conservation for the schemes and demonstrate 1D radiative-shock tests where LIMEX methods significantly outperform traditional Lie-Trotter operator splitting in both accuracy and error constants. The approach offers robust, higher-order time integration for multiphysics RHD problems and points to future extensions to more complex radiative transport and adaptive, NPRK-based schemes.

Abstract

Radiation hydrodynamics are a challenging multiscale and multiphysics set of equations. To capture the relevant physics of interest, one typically must time step on the hydrodynamics timescale, making explicit integration the obvious choice. On the other hand, the coupled radiation equations have a scaling such that implicit integration is effectively necessary in non-relativistic regimes. A first-order Lie-Trotter-like operator split is the most common time integration scheme used in practice, alternating between an explicit hydrodynamics step and an implicit radiation solve and energy deposition step. However, such a scheme is limited to first-order accuracy, and nonlinear coupling between the radiation and hydrodynamics equations makes a more general additive partitioning of the equations non-trivial. Here, we develop a new formulation and partitioning of radiation hydrodynamics with gray diffusion that allows us to apply (linearly) implicit-explicit Runge-Kutta time integration schemes. We prove conservation of total energy in the new framework, and demonstrate 2nd-order convergence in time on multiple radiative shock problems, achieving error 3--5 orders of magnitude smaller than the first-order Lie-Trotter operator split at the hydrodynamic CFL, even when Lie-Trotter applies a 3rd-order TVD Runge-Kutta scheme to the hydrodynamics equations.

Figures (7)

• Figure 1: Mach-1.2 density profiles for $t\in\{0,0.25,0.5,0.75\}$ns
• Figure 4: Mach-1.2 internal energy profiles for $t\in\{0,0.25,0.5,0.75\}$ns
• Figure 7: Mach-1.2 relative $\ell^2$-error; dotted lines demonstrate first, second, and third order convergence.
• Figure 8: Mach-3 relative $\ell^1$-error; dotted lines demonstrate first, second, and third order convergence.
• Figure 9: Mach-45 relative $\ell^\infty$ error; dotted lines demonstrate first, second, and third order convergence.