Second-order discretization in space and time for radiation-hydrodynamics

Second-order accurate discretizations for radiation-hydrodynamics are currently an area of interest in the high energy density laboratory physics and astrophysics communities. Second-order methods used to solve the hydrodynamics equations and second-order methods used to solve the radiation transport equation often differ fundamentally, making it difficult to combine them in a second-order manner. Here, we present an implicit–explicit (IMEX) method for solving the 1D equations of radiation-hydrodynamics that is second-order accurate in space and time. Our radiation-hydrodynamics model consists of the 1D Euler equations coupled with a gray radiation S₂ approximation. Our RH method combines the MUSCL-Hancock method for solving the Euler equations with the TR/BDF2 time integration scheme and the linear-discontinuous Galerkin finite-element spatial discretization scheme for the S₂ radiation equations. The MUSCL-Hancock method is commonly used for hydrodynamic calculations and the linear-discontinuous Galerkin scheme is the standard for the S_n equations of radiative transfer. While somewhat similar, these schemes vary fundamentally with respect to the treatment of spatial slopes. We address the challenges inherent to coupling these different numerical methods and demonstrate how these challenges can 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.