High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study (Communications in Computational Physics, 2018)
Fixed-point iterative sweeping methods were developed in the literature toefficiently solve static Hamilton-Jacobi equations. This class of methods utilizes theGauss-Seidel iterations and alternating sweeping strategy to achieve fast convergencerate. They take advantage of the properties of hyperbolic partial differential equations(PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobiequation in a certain direction simultaneously in each sweeping order. Differentfrom other fast sweeping methods, fixed-point iterative sweeping methods have theadvantages such as that they have explicit forms and do not involve inverse operationof nonlinear local systems. In principle, it can be applied to solving very generalequations using any monotone numerical fluxes and high order approximations easily.In this paper, based on the recently developed fifth order WENO schemes which improvethe convergence of the classical WENO schemes by removing slight post-shockoscillations, we design fifth order fixed-point sweeping WENO methods for efficientcomputation of steady state solution of hyperbolic conservation laws. Especially, weshow that although the methods do not have linear computational complexity, theyconverge to steady state solutions much faster than regular time-marching approachby stability improvement for high order schemes with a forward Euler time-marching.