TY - JOUR
T1 - Fixed-Point Fast Sweeping WENO Methods for Steady State Solution of Scalar Hyperbolic Conservation Law 
AU - S. Chen
JO - International Journal of Numerical Analysis and Modeling
VL - 1
SP - 117
EP - 130
PY - 2014
DA - 2014/11
SN - 11
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/517.html
KW - fast sweeping methods, WENO methods, Jacobi iteration, Gauss-Seidel iteration, hyperbolic conservation laws, steady state.
AB - <p style="text-align: justify;">Fast sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilize the Gauss-Seidel iterations and alternating 
sweeping strategy to achieve fast convergence rate. They take advantage of the properties
of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of
the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping 
order. In [16], the Gauss-Seidel idea and alternating sweeping strategy were adopted to the
time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations, and numerical 
examples verified at least a 2 times acceleration of convergence even on relatively coarse
grids. In this paper, we apply the same approach to solve steady state solution of hyperbolic
conservation laws. We use numerical examples to verify that a 2 times acceleration of convergence
is achieved. And the computational cost is exactly the same as the time-marching scheme at each
iteration. Based on the Gauss-Seidel iterations, we explore the successive overrelaxation (SOR)
approach to further improve the performance of our fixed-point sweeping methods.</p>