High Order Strong Stability Preserving Time Integrators and Numerical Wave Propagation Methods for Hyperbolic PDEs

Abstract

Hyperbolic PDEs describe the great variety of physical phenomena governed by wave behavior. Numerical methods are necessary to provide approximate solutions to real wave propagation problems. High order accurate methods are often essential to achieve accurate solutions on computationally feasible grids. However, maintaining numerical stability and satisfying physical constraints becomes increasingly difficult as higher order methods are employed. In this thesis, high order numerical tools for hyperbolic PDEs are developed in a method of fines framework. Optimal high order accurate strong stability preserving (SSP) time integrators, for both linear and nonlinear systems, are developed. Improved SSP methods are found as a result of rewriting the relevant optimization problems in a form more amenable to efficient solution. A new, very general class of low-storage Runge-Kutta methods is proposed (based on the form of some optimal SSP methods) and shown to include methods with properties that cannot be achieved by existing classes of low-storage methods. A high order accurate semi-discrete wave propagation method is developed in one and two dimensions using wave propagation Riemann solvers and high order weighted essentially non-oscillatory (WENO) reconstruction. The space and time discretizations are combined to achieve a high order accurate method for general hyperbolic PDEs. This method is applied to model solitary waves in a nonlinear, non-dispersive periodic heterogeneous medium.