A Godunov Method for Multidimensional Radiation Magnetohydrodynamics based on a variable Eddington tensor

We describe a numerical algorithm to integrate the equations of radiation magnetohydrodynamics in multidimensions using Godunov methods. This algorithm solves the radiation moment equations in the mixed frame, without invoking any diffusion-like approximations. The moment equations are closed using a variable Eddington tensor whose components are calculated from a formal solution of the transfer equation at a large number of angles using the method of short characteristics. We use a comprehensive test suite to verify the algorithm, including convergence tests of radiation-modified linear acoustic and magnetosonic waves, the structure of radiation modified shocks, and two-dimensional tests of photon bubble instability and the ablation of dense clouds by an intense radiation field. These tests cover a very wide range of regimes, including both optically thick and thin flows, and ratios of the radiation to gas pressure of at least 10^{-4} to 10^{4}. Across most of the parameter space, we find the method is accurate. However, the tests also reveal there are regimes where the method needs improvement, for example when both the radiation pressure and absorption opacity are very large. We suggest modifications to the algorithm that will improve accuracy in this case. We discuss the advantages of this method over those based on flux-limited diffusion. In particular, we find the method is not only substantially more accurate, but often no more expensive than the diffusion approximation for our intended applications.

💡 Research Summary

This paper presents a multidimensional radiation magnetohydrodynamics (RMHD) solver that integrates the Godunov finite‑volume framework with a variable Eddington tensor (VET) closure. Unlike the widely used flux‑limited diffusion (FLD) approach, which imposes an artificial limit on the radiation flux and can produce non‑physical propagation speeds in optically thin regions, the authors solve the radiation moment equations directly in the mixed frame and close them with a VET computed from a formal solution of the transfer equation. The VET is obtained by the short‑characteristics method, which evaluates the specific intensity along many discrete angles, thereby capturing anisotropic radiation fields and non‑linear radiation‑matter coupling without diffusion‑type approximations.

The numerical scheme combines a high‑order Godunov Riemann solver for the MHD subsystem with a second‑order TVD Runge‑Kutta time integrator. Spatial reconstruction uses a limited MUSCL approach to achieve third‑order accuracy in smooth regions, while magnetic field divergence is controlled via constrained transport. Radiation source terms are added to the conserved variables, and the VET is recomputed each hydrodynamic step, ensuring consistent coupling between radiation and fluid dynamics.

A comprehensive test suite validates the algorithm across a vast parameter space: linear acoustic and magnetosonic waves modified by radiation, radiation‑modified shock structures, the two‑dimensional photon‑bubble instability, and the ablation of dense clouds by intense radiation. These tests span optical depths from τ ≪ 1 to τ ≫ 1 and radiation‑to‑gas pressure ratios from 10⁻⁴ to 10⁴. In most regimes the method reproduces analytical solutions and converges at second order, with errors typically below a few percent. Compared with FLD, the VET‑Godunov scheme yields markedly better phase accuracy for wave propagation and captures shock precursors and anisotropic fluxes that FLD smears out.

The authors identify a limitation: when both radiation pressure and absorption opacity become extremely large, the VET iteration can become numerically unstable, leading to spurious transients. They propose two remedies: increasing the number of fixed‑point iterations for the VET calculation, or employing a hybrid strategy that reverts to a diffusion approximation in highly opaque cells while retaining the VET elsewhere.

Overall, the paper demonstrates that a Godunov‑based RMHD solver with a variable Eddington tensor can achieve high accuracy without a substantial increase in computational cost. The method is particularly well‑suited for astrophysical problems where radiation and magnetic fields interact strongly, such as supernova explosions, radiation‑pressure‑dominated accretion disks, and high‑energy plasma environments.