A fifth-order absolutely convergent fixed-point fast sweeping hybrid alternative WENO scheme for steady state of hyperbolic conservation laws

TL;DR

This paper introduces a fifth-order hybrid fast sweeping scheme for steady hyperbolic conservation laws that achieves absolute convergence more efficiently than previous methods and popular time-marching approaches.

Contribution

It develops a novel fifth-order hybrid fixed-point fast sweeping WENO scheme with unequal-sized substencils for improved efficiency and convergence in steady state solutions.

Findings

Achieves absolute convergence to machine zero residuals.

More efficient than third-order TVD Runge-Kutta methods.

Enhances flexibility with arbitrary monotone numerical fluxes.

Abstract

In this paper, we extend the previous work on absolutely convergent fixed-point fast sweeping WENO methods by Li et al. (J. Comput. Phys. 443: 110516, 2021) and design a fifth-order hybrid fast sweeping scheme for solving steady state problems of hyperbolic conservation laws. Unlike many other fast sweeping methods, the explicit property of fixed-point fast sweeping methods provides flexibility to apply the alternative weighted essentially non-oscillatory (AWENO) scheme with unequal-sized substencils as the local solver, which facilitates the usage of arbitrary monotone numerical fluxes. Furthermore, a novel hybrid technique is designed in the local solver to combine the nonlinear AWENO interpolation with the linear scheme for an additional improvement in efficiency of the high-order fast sweeping iterations. Numerical examples show that the developed fixed-point fast sweeping hybrid AWENO method with unequal-sized substencils can achieve absolute convergence (i.e., the residue of the fast sweeping iterations converges to machine zero / round off errors) more easily than the original AWENO method with equal-sized substencils, and is more efficient than the popular third-order total variation diminishing (TVD) Runge-Kutta time-marching approach to converge to steady state solutions.