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

TL;DR

The paper targets efficient, high-order steady-state solvers for hyperbolic conservation laws. It develops a fifth-order hybrid AWENO scheme with unequal-sized substencils within a fixed-point fast sweeping framework, enabling direct use of monotone fluxes while maintaining absolute convergence. A novel hybrid interpolation combines nonlinear AWENO with a linear upwind reconstruction, guided by a simple quadratic-troubled-cell test, to boost efficiency. Numerical experiments across diverse benchmarks demonstrate robust absolute convergence, flux-flexibility, and substantial speedups over traditional RK time-marching methods, highlighting practical impact for CFD and related applications.

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.

Figures (10)

• Figure 1: Example 2: Regular shock reflection. 30 equally spaced density contours from 1.1 to 2.6 of the steady states of numerical solutions by different iterative schemes, and the identified troubled-cells where WENO interpolation is used in the FS-HAUSWENO scheme.
• Figure 2: Example 2: Regular shock reflection. The convergence history of the residue as a function of number of iterations for different schemes with various CFL numbers.
• Figure 3: Example 3: Supersonic flow past a plate with an attack angle. 30 equally spaced pressure contour from 0.04 to 0.17 of the converged steady states of numerical solutions by different iterative schemes, and the identified troubled-cells where the WENO interpolation is used in the FS-HAUSWENO scheme.
• Figure 4: Example 3: Supersonic flow past a plate with an attack angle. The convergence history of the residue as a function of number of iterations for different schemes with various CFL numbers.
• Figure 5: Example 4: Flow around a circular cylinder. 30 equally spaced pressure contour from 0.2 to 12 of the converged steady states of numerical solutions by different iterative schemes, and the identified troubled-cells where the WENO interpolation is used in the FS-HAUSWENO scheme.