# A direct Eulerian GRP scheme for radiation hydrodynamical equations in diffusion limit

## TL;DR

This paper introduces a second-order accurate Eulerian GRP scheme for radiation hydrodynamical equations in the diffusion limit, effectively handling nonlinear waves and discontinuities without explicit flux expressions.

## Contribution

It develops a novel direct Eulerian GRP scheme that analytically resolves nonlinear waves using generalized Riemann invariants and jump conditions, improving accuracy in radiation hydrodynamics.

## Key findings

- Achieves second-order accuracy in numerical tests
- Effectively captures strong discontinuities
- Demonstrates high resolution of nonlinear waves

## Abstract

The paper proposes a second-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for the radiation hydrodynamical equations (RHE) in the zero diffusion limit. The difficulty comes from no explicit expression of the flux in terms of the conservative vector. The characteristic fields and the relations between the left and right states across the elementary-waves are first studied, and then the solution of the one-dimensional Riemann problem is analyzed and given. Based on those, the direct Eulerian GRP scheme is derived by directly using the generalized Riemann invariants and the Runkine-Hugoniot jump conditions to analytically resolve the left and right nonlinear waves of the local GRP in the Eulerian formulation. Several numerical examples show that the GRP scheme can achieve second-order accuracy and high resolution of strong discontinuity.