Mathematical Models and Numerical Methods for Radiation Hydrodynamics

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We present mathematical models and numerical methods for radiation hydrodynamics. The mathematical models range from compressible Euler to incompressible Navier-Stokes problems for the hydrodynamic part and radiative heat transfer and simplified PN-approximations for the radiation part. The coupling between the hydrodynamical flow and the radiative signal is carried out in two different approaches for the radiative stage. The first approach is based on asymptotic analysis while the second approach is based on entropy principle. The mathematical aspects and the physical features are studied for both approaches by considering experiments in two and three dimensional problems. To develop accurate and efficient solvers for radiation hydrodynamics, we propose two class of numerical procedures: The first class consists of high-order relaxation schemes for the hydrodynamic equations. The main advantage of these schemes lies on the semilinear structure of the relaxation systems which can be solved without using either Riemann solvers or nonlinear iterations. Special attention is given to a third order non-oscillatory reconstruction in space and a TVD implicit-explicit integration in time. Numerical results are shown for a wide variety of benchmark tests from computational fluid dynamies. The second class of numerical methods deals with efficient solvers for radiative heat transfer problems. We propose a family of multilevel algorithms for both radiative transfer and heat conduction using the same mesh hierarchy. The Atkinson-Brakhage approximate inverse is used as a smoother while a Newton-Krylov method solves the coarse problems. The Robustness and the effectiveness of the multilevel algorithms are shown using several test problems on grey and frequency-coupled media. Comparisons to other numerical methods from the literature are also included in this work.