High order schemes: Convergence for hyperbolic conservation laws and applications in computational cosmology

This thesis contains three related topics on the high order numerical methods for hyperbolic equations.;In the first part, we prove the convergence of up to third order Godunov-type schemes toward the entropy solution of hyperbolic conservation laws under large time steps. Practical implementation of the scheme for 1-D scalar convex conservation laws is discussed. Numerical examples are demonstrated to show the applicability of these numerical schemes and to verify the theoretical convergence result as well as the correct order of accuracy.;In the second part, we investigate the issue of convergence toward entropy solutions for high order numerical methods approximating conservation laws. It is showed that convergence may fail for certain nonconvex conservation laws. A first order monotone modification and a second order modification with an entropic projection in nonconvex regions are proposed to enforce the convergence toward the entropy solution for general nonconvex conservation laws.;The third topic concerns the application of the fifth order finite difference weighted essentially non-oscillatory (WENO) scheme on radiative transfer equations in computational cosmology. Numerical simulation results are demonstrated with corresponding physical explanations.