Algorithms for hyperbolic balance laws with multiscale behavior: Applications in radiation hydrodynamics

Many physical systems related to hydrodynamics, combustion, and plasma physics are characterized by a continuum approximation. However, due to their complicated nature one must resolve many of these systems computationally. Some mathematical properties associated with hyperbolic systems are inherent to such physical phenomena and allow one to construct general numerical techniques. This thesis focuses on one family of finite volume schemes, the Godunov method and its higher-order extensions via predictor-corrector techniques.;From a mathematical perspective, radiation hydrodynamics, which couples advection-reaction processes related to matter and radiation, can be thought of as a system of hyperbolic balance laws with dual multiscale behavior (multiscale behavior associated with the hyperbolic wave speeds c/ainfinity ∼ 106 where c is the speed of light and ainfinity is the speed of sound as well as multiscale behavior associated with source term relaxation S/ainfinity ∼ [0, 106] where S is the magnitude of the source term). However, these behaviors are quite different and cause breakdowns in monotonicity and stability, respectively, all the while influencing the temporal resolution Deltat of the problem. Given such complexity, a numerical technique must be coarsely gridded, steady state preserving, and asymptotic preserving. A hybrid method is proposed that combines a backward Euler upwinding scheme for the radiation components (treating the first multiscale behavior) with a modified Godunov scheme for the material components (treating the second multiscale behavior).;The modified Godunov scheme, which is composed of a predictor step that is based on Duhamels principle and a corrector step that is based on Picard iteration, is first applied to an inviscid hydrodynamical system that is defined by the Euler equations and subjected to a stiff energy source term. Such a setup causes the system to transition from adiabatic to isothermal behavior in the limit of an infinite cooling rate. Analytic investigation and numerical tests examine the dynamical scaling related to S/ainfinity and show the scheme to be second-order accurate, unsplit such that it directly couples stiff source term effects to the hyperbolic structure of the system of balance laws, explicit on the material flow scale via a CFL condition, asymptotic preserving, stable, robust, and uniformly well behaved across a range of parameters.;In order to examine the dynamical scaling related to c/a infinity, the modified Godunov scheme is applied to the radiation subsystem, a simpler system that is related to radiation hydrodynamics and maintains essential hyperbolic-parabolic behaviors. Analytic investigation and numerical tests show the modified Godunov scheme to be second-order accurate, unsplit, asymptotic preserving, and uniformly well behaved from the photon free streaming (hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and to the strong equilibrium diffusion (hyperbolic) limit. Yet, temporally evolving the radiation subsystem according to a CFL condition that is explicit on the speed of light illustrates the multiscale nature associated with the hyperbolic wave speeds and need for a hybrid approach which implicitly advances the radiation components.;The hybrid Godunov method combines a backward Euler upwinding scheme with the modified Godunov scheme to evolve full radiation hydrodynamical problems in one spatial dimension. This backward Euler scheme is first-order accurate, uses an implicit HLLE flux function to temporally advance the radiation components according to the material flow scale, has a consistent discretization, and recovers diffusion behavior even for large cell-optical depths. For reasons related to monotonicity preserving conditions and the stiffness associated with some hyperbolic waves moving at the speed of light, one is forced to use a first-order backward Euler scheme which is unconditionally TVD stable. Analytic investigation and numerical tests focusing on the material components, radiation components, and fully coupled system show this hybrid approach to be uniformly well behaved across various parameter regimes with some asymptotic preserving properties. The advantages of this algorithm is that it is accurate, stable, robust, explicit on the material flow scale, and fully couples matter and radiation without invoking a diffusion-type approximation. Algorithmic details are given for extending the hybrid Godunov method to multiple spatial dimensions.