High Order Fast Iterative Methods for Steady State of Hyperbolic Partial Differential Equations

Fast sweeping methods are a class of efficient iterative methods for solving steady state hyperbolic partial differential equations (PDEs). This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic PDEs and try to cover a family of characteristics of the corresponding equation in a certain direction simultaneously in each sweeping order. The first order fast sweeping method for solving Eikonal equations has linear computational complexity, namely, the computational cost is O(N) where N is the number of grid points of the computational mesh. Recently, a second order fast sweeping method with linear computational complexity was developed. The method is based on a discontinuous Galerkin (DG) finite element solver and causality indicators which guide the information flow directions of the nonlinear Eikonal equations. In the first part of this dissertation, we extend previous work and develop a third order fast sweeping method with linear computational complexity for solving Eikonal equations. A novel approach is designed for capturing the causality information in the third order DG local solver.

Fixed-point iterative sweeping methods were also developed in the literature to efficiently solve static Hamilton-Jacobi equations. Different from other fast sweeping methods, fixed-point iterative sweeping methods have the advantages such as having explicit forms and not involving inverse operation of nonlinear local systems. In principle, they can be applied in solving very general equations using any monotone numerical fluxes and high order approximations easily. In the second part of this dissertation, based on the recently developed fifth order weighted essentially non-oscillatory (WENO) schemes which improve the convergence of the classical WENO schemes by removing slight post-shock oscillations, we design fifth order fixed-point sweeping WENO methods for efficient computation of steady state solution of hyperbolic conservation laws. Especially, we show that although the methods do not have linear computational complexity, they converge to steady state solutions much faster than regular time-marching approach by stability improvement for high order schemes with a forward Euler time-marching.