The Efficient Schemes For A Class Of Radiant Hydrodynamics Equations

The nuclear fusion energy will be the important energy for the sur-vival and development of human. The inertial confinement fusion (ICF) is an important way to carry out the thermonuclear fusion, and the radiant hydrodynamics equations (RHEs) is an important mathematical model to describe the implosion dynamics process in ICF. The decoupling method is a main method for computing the RHEs, and its basic idea is to convert RHEs essentially to the radiant fluid dynamics equations and the radiant heat conduction equations, which can be solved respectively. However, there exist many difficulties in their numerical solution due to their particular fea-tures such as multi-scale, strong nonlinearity, strong discontinuity and high density ratio etc.. The main contents of this thesis are as follows:Two kinds of common difference schemes on the uniform grids are im-proved by using the modified coefficient method, and the MCupwind scheme and the MCENO scheme are obtained. Furthermore, we consider the sec-ond order finite difference ENO scheme on non-uniform grids, and modify the coefficients of the spatial discretized terms, to improve the accuracy and maintain the stability, the NU-MCENO scheme is obtained. The numerical results show that the NU-MCENO scheme is more efficient than the corre-sponding ENO scheme.Based on the FDWENO scheme and the RKDG scheme, a hybrid scheme is constructed, which have the advantages including the fast computation velocity of the FDWENO scheme and the adaptability to arbitrary grids of the RKDG scheme. The basic idea is to separate the computing domain to several sub-domains, and the FDWENO scheme and the RKDG scheme are used in the different sub-domains, respectively. The key is how to deal with the numerical flux on the public boundaries of the sub-domains. For this purpose, we give an approach and analyze the conservative error. The results from numerical experiments show that this hybrid scheme is reliable and efficient. The error asymptotic expansions and super convergence of an important finite volume element scheme are studied. First, for the stationary diffusion problems, the point-wise asymptotic expansions and super convergence of the iso-parametric bilinear finite volume element solution are proved on the uniform grids with small disturbance. Second, for the linear parabolic prob-lems, the point-wise asymptotic expansions and super convergence of the iso-parametric bilinear finite volume element solution are firstly proved on the uniform grids. The numerical results confirm the obtained theoretical results.For the Rayleigh-Taylor instability problems from the laser ablation, the FDWENO scheme and the hybrid scheme are used to compute the radiant fluid dynamics equations obtained by decoupling respectively, and the iso-parametric bilinear finite volume element scheme is used to compute the radiant heat conduction equations obtained by decoupling. The numerical results show that the schemes can preserve the symmetry of physical quantity, and the energy conservative error is small. Hence the reliability and validity of the new schemes are verified.