Radiative shock test

This test problem demonstrates the correct coupled solution of the hydrodynamics and radiation moment equations for a subcritical radiative shock. The steady-state solution in the nonequilibrium radiation diffusion approximation is given by a set of coupled ODEs that can be solved to arbitrary precision following the method of (Lowrie & Edwards, 2008)1.

Parameters

The dimensionless shock parameters (Lowrie & Edwards, 2008)1 are:

\[\begin{aligned} P_0 = 1.0 \times 10^{-4} \\ \sigma_a = 1.0 \times 10^{6} \\ \mathcal{M}_0 = 3.0 \\ \gamma = 5/3 \\ \end{aligned}\]

Following (Skinner, Dolence, Burrows, Radice, & Vartanyan, 2019)2, we scale to dimensional values assuming

\[\begin{aligned} \mu = m_H \\ c_v = \frac{k_B}{\mu (\gamma - 1)} \, \text{erg} \, \text{g}^{-1} \, \text{K}^{-1} \\ c_{s,0} = 1.73 \times 10^{7} \, \text{cm} \, \text{s}^{-1} \\ \kappa = 577.0 \, \text{cm}^{-1} \\ \end{aligned}\]

and obtain the following pre-shock and post-shock states:

\[\begin{aligned} T_0 = 2.18 \times 10^6 \, \text{K} \\ \rho_0 = 5.69 \, \text{g} \, \text{cm}^{-3} \\ v_0 = 5.19 \times 10^7 \, \text{cm} \, \text{s}^{-1} \\ T_1 = 7.98\times 10^6 \, \text{K} \\ \rho_1 = 17.1 \, \text{g} \, \text{cm}^{-3} \\ v_1 = 1.73 \times 10^7 \, \text{cm} \, \text{s}^{-1} \, . \end{aligned}\]

We adopt a reduced speed of light (as used in (Skinner, Dolence, Burrows, Radice, & Vartanyan, 2019)2)

\[\hat c = 10 (v_0 + c_{s,0}) \, .\]

Solution

Since the solution is given assuming radiation diffusion, we set the Eddington factor (as used in the Riemann solver for the radiation moment equations) to a constant value of \(1/3\) everywhere.

We use the RK2 integrator with a CFL number of 0.2 and a mesh of 256 equally-spaced zones. After 3 shock crossing times, we obtain a solution for the radiation temperature and matter temperature that agrees to better than 0.5% (in relative L1 norm) with the steady-state ODE solution to the radiation hydrodynamics equations:

The radiation temperature is shown in the black solid and dashed lines, with the dashed line showing the semi-analytic solution. The material temperature is shown in the red lines, with the semi-analytic solution shown with the dashed line.

  1. Lowrie, R. B., & Edwards, J. D. (2008). Radiative shock solutions with grey nonequilibrium diffusion. Shock Waves, 18(2), 129–143. https://doi.org/10.1007/s00193-008-0143-0 ↩↩

  2. Skinner, M. A., Dolence, J. C., Burrows, A., Radice, D., & Vartanyan, D. (2019). FORNAX: A Flexible Code for Multiphysics Astrophysical Simulations. 241(1), 7. https://doi.org/10.3847/1538-4365/ab007f ↩↩