  Fixed-point iterative sweeping methods were developed in the literature to
efficiently solve steady state solutions of Hamilton-Jacobi equations and
hyperbolic conservation laws. Similar as other fast sweeping schemes, the key
components of this class of methods are the Gauss-Seidel iterations and
alternating sweeping strategy to achieve fast convergence rate. Furthermore,
good properties of fixed-point iterative sweeping methods include that they
have explicit forms and do not involve inverse operation of nonlinear local
systems, and they can be applied to general hyperbolic equations using any
monotone numerical fluxes and high order approximations easily. In [L. Wu,
Y.-T. Zhang, S. Zhang and C.-W. Shu, Commun. Comput. Phys., 20 (2016)], a fifth
order fixed-point sweeping WENO scheme was designed and it was shown that the
scheme converges much faster than the total variation diminishing (TVD)
Runge-Kutta approach by stability improvement of high order schemes with a
forward Euler time-marching. An open problem is that for some benchmark
numerical examples, the iteration residue of the fixed-point sweeping WENO
scheme hangs at a truncation error level instead of settling down to machine
zero. This issue makes it difficult to determine the convergence criterion for
the iteration and challenging to apply the method to complex problems. To solve
this issue, in this paper we apply the multi-resolution WENO scheme developed
in [J. Zhu and C.-W. Shu, J. Comput. Phys., 375 (2018)] to the fifth order
fixed-point sweeping WENO scheme and obtain an absolutely convergent
fixed-point fast sweeping method for steady state of hyperbolic conservation
laws, i.e., the residue of the fast sweeping iterations converges to machine
zero / round off errors for all benchmark problems tested.
