On Consistent Time-Integration Methods for Radiation Hydrodynamics in the Equilibrium Diffusion Limit: Low-Energy-Density Regime

We compare the accuracy of three, mixed explicit–implicit schemes for simulating nonrelativistic, radiative hydrodynamic phenomena in the equilibrium diffusion limit. Only the “low-energy-density” regime is considered, where it is possible to ignore the effects of radiation pressure and energy density in comparison to the fluid pressure and energy density. The governing equations are then those of compressible Eulerian hydrodynamics with a nonlinear, radiative heat-transfer term appearing in the energy equation. All three finite-volume methods in this study utilize an explicit Godunov method with an approximate Riemann solver to integrate the Euler equations, but differ in their iterative treatment of the radiation diffusion term, which is handled in an “operator-split” fashion. In the first method, diffusive effects are computed with a linearized implicit technique that does not converge nonlinearities within a computational time step. In the other two methods, a Jacobian-free Newton–Krylov procedure is used to converge the nonlinearities, and improved accuracy (but not always greater efficiency) is achieved over the more traditional linearized–implicit approach. The two Newton–Krylov methods differ in their order of accuracy in time; one is strictly first-order accurate, while the other attempts to achieve second-order accuracy by making use of a predictor–corrector architecture. Several examples are considered to demonstrate the convergence properties of the three schemes, but attention is limited to spherically symmetric problems such as the one-dimensional point explosion.