High order well-balanced numerical schemes for hyperbolic systems with source terms

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source term. This thesis contains several topics on constructing genuinely high order accurate well balanced numerical schemes, which can preserve exactly these steady state solutions.;In the first part, we start our investigation by designing high order well balanced WENO finite difference schemes for the still water solution of the shallow water equations, and then generalize our idea to a general class of balance laws with separable source terms. Well balanced high order finite volume weighted essentially non-oscillatory (WENO) schemes and Runge-Kutta discontinuous Galerkin (RKDG) finite element schemes, which are more suitable for computations in complex geometry and/or for using adaptive meshes, are also designed for the same class of balance laws. The key ingredient in our design is a special decomposition of the source term before discretization, which allows us to design specific approximations such that the resulting schemes satisfy the well balanced property, and at the same time maintain their original high order accuracy and essentially non-oscillatory property for general solutions.;In the second part, we present a different approach to design high order well-balanced finite volume WENO schemes and RKDG finite element methods. We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly, if a small modification on either the initial condition or the flux is provided. The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method.;The third topic is related to the moving steady state solution of the shallow water equation, which cannot be preserved by the above methods. We introduce a new technique to obtain high order finite volume schemes for this problem, based on a special treatment of the flux and source term. Extensive numerical simulations are performed to verify high order accuracy, the well balanced property, and good resolution for smooth and discontinuous solutions.