Second-Order Accurate Method for Solving Radiation-Hydrodynamics

Second-order discretization for radiation-hydrodynamics is currently an area of great interest. Second-order methods used to solve the respective single-physics problems often differ fundamentally, making it difficult to combine them in a second- order manner. Here, we present a method for solving the equations of radiation hydrodynamics that is second-order accurate in space and time. We achieve this accuracy by combining modern methods used in standard single-physics calculations. This method is defined for a 1-D model of compressible fluid dynamics coupled with grey radiation diffusion and combines the MUSCL-Hancock method for solving the Euler equations with the TR/BDF2 scheme in time and a linear-discontinuous finite-element method in space for solving the equations of radiative transfer. Though uncommon for radiation diffusion calculations, the linear-discontinuous method is a standard for radiation transport applications. We address the challenges inherent to using different spatial discretizations for the hydrodynamics and radiation components and demonstrate how these may be overcome. Using the method of manufactured solutions, we show that the method is second-order accurate in space and time for both the equilibrium diffusion and streaming limit, and we show that the method is capable of computing radiative shock solutions accurately by comparing our results with semi-analytic solutions