[Paper Review] A radiation-hydrodynamics scheme valid from the transport to the diffusion limit

E. Audit, Pierre Charrier+2 more|arXiv (Cornell University)|Jun 17, 2002

Computational Fluid Dynamics and AerodynamicsEngineering12 references21 citations

TL;DR

This paper presents a novel radiation-hydrodynamics scheme based on the M₁ closure model that accurately captures both the transport (free-streaming) and diffusion limits in radiative transfer. By asymptotic-preserving modification of the HLLE Riemann solver and coupling with a moving-grid, time-implicit framework, the method maintains robustness and accuracy across extreme opacity gradients, as validated in radiative shock and protostar formation simulations with high-resolution resolution of sharp shocks and diffusion fronts.

ABSTRACT

We present in this paper the numerical treatment of the coupling between hydrodynamics and radiative transfer. The fluid is modeled by classical conservation laws (mass, momentum and energy) and the radiation by the grey moment $M_1$ system. The scheme introduced is able to compute accurate numerical solution over a broad class of regimes from the transport to the diffusive limits. We propose an asymptotic preserving modification of the HLLE scheme in order to treat correctly the diffusion limit. Several numerical results are presented, which show that this approach is robust and have the correct behavior in both the diffusive and free-streaming limits. In the last numerical example we test this approach on a complex physical case by considering the collapse of a gas cloud leading to a proto-stellar structure which, among other features, exhibits very steep opacity gradients.

Motivation & Objective

• To develop a single numerical scheme that accurately models radiation-hydrodynamics across all regimes, from free-streaming to diffusion, without domain decomposition.
• To overcome the limitations of separate models for different regimes, which introduce errors at transition zones.
• To ensure correct behavior in both the streaming and diffusion limits, especially in problems with strong opacity gradients.
• To enable high-resolution simulations of complex astrophysical phenomena like protostar formation with accurate treatment of radiative shocks.
• To implement the scheme within a moving-grid, time-implicit framework to handle disparate time and spatial scales.

Proposed method

• The fluid is modeled using classical conservation laws for mass, momentum, and energy.
• Radiation is described by the grey M₁ moment system, which uses the minimum entropy principle to close the moment equations and capture anisotropy.
• The Riemann solver is constructed by decoupling the hydrodynamic and radiation subsystems, solving them separately under the assumption of negligible coupling and source terms.
• Non-conservative coupling terms are reintroduced by using the hydrodynamic velocity in the radiation equations, improving accuracy in the streaming regime.
• The HLLE scheme is asymptotically preserved by modifying it to handle the stiff relaxation term in the radiation subsystem, ensuring correct diffusion limit behavior.
• The full solver is implemented in a moving-grid, time-implicit framework to manage large disparities in time and spatial scales in astrophysical simulations.

Experimental results

Research questions

• RQ1 Can a single numerical scheme accurately simulate radiation-hydrodynamics across the entire range of regimes, from free-streaming to diffusion, without domain decomposition?
• RQ2 How can a Godunov-type scheme be modified to preserve the diffusion limit while maintaining accuracy in the streaming regime?
• RQ3 What is the impact of high spatial resolution and adaptive grids on resolving sharp radiative shocks and opacity gradients in protostar formation?
• RQ4 How does the M₁ closure model perform in capturing the transition between diffusion and free-streaming regimes in realistic astrophysical flows?
• RQ5 Can the proposed scheme maintain stability and accuracy in long-term simulations of collapsing gas clouds with strong luminosity and temperature gradients?

Key findings

• The scheme successfully captures the correct physical behavior in both the free-streaming and diffusion limits, with no artificial transition errors.
• In the radiative shock test, the method resolves a sharp temperature spike at the shock with high spatial resolution, confirming its ability to model supercritical shocks.
• The luminosity profile shows a sharp jump at the accretion shock, consistent with energy dissipation and radiative cooling, while the radiation temperature varies smoothly.
• The Eddington factor drops to one-third in the central core, confirming the diffusion regime, and increases toward the shock before decreasing again at large radii, reflecting the transition to free-streaming.
• The grid resolution is dynamically adapted, achieving high resolution at the shock and central core, with spatial resolution varying over 10 decades to capture steep gradients.
• The simulation of protostar formation shows a density contrast exceeding 18 decades and a velocity profile indicating both expanding outer regions and collapsing inner envelopes, with a well-resolved accretion shock.