Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

Abstract

A new type of high-order multi-resolution weighted essentially non-oscillatory (WENO) schemes (Zhu and Shu in J Comput Phys, 375: 659-683, 2018) is applied to solve for steady-state problems on structured meshes. Since the classical WENO schemes (Jiang and Shu in J Comput Phys, 126: 202-228, 1996) might suffer from slight post-shock oscillations (which are responsible for the residue to hang at a truncation error level), this new type of high-order finite-difference and finite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state simulations. This new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes, could obtain fifth-order, seventh-order, and ninth-order in smooth regions, and could gradually degrade to first-order so as to suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO schemes can be any positive numbers on the condition that their sum is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finitedifference and finite-volume WENO schemes for solving steady-state problems. In comparison with the classical fifth-order finite-difference and finite-volume WENO schemes, the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems.

Absolutely convergent fixed-point fast sweeping WENO methods for steady state of hyperbolic conservation laws

“…WENO schemes with unequal-sized sub-stencils [35,39] have shown nice property in convergence to steady states. In this paper, we adopt the fifth order multi-resolution WENO scheme [37] for the spatial discretization of (2.1) in order to achieve an absolutely convergent fixed-point fast sweeping method for steady state of hyperbolic conservation laws.…”

“…It is a popular method to march numerical solutions of high order WENO schemes to steady states, see e.g. [11,39]. In order to achieve faster convergence to steady state solutions of high order WENO schemes for solving hyperbolic PDEs than the Jacobi type fixed-point iterations, one approach is to apply the fast sweeping techniques so that the important characteristics property of hyperbolic PDEs can be utilized in the iterations.…”

“…In this section we perform numerical experiments to test the performance of the proposed fifth order absolutely convergent fixed-point fast sweeping WENO scheme for solving steady-state problems. These numerical examples come from the literature about numerical studies on steady state solutions of hyperbolic conservation laws, e.g., [24,36,39]. Computational efficiency of the fast sweeping scheme and the other two iterative schemes discussed in the last section is compared.…”

“…The WENO schemes on unequal-sized sub-stencils exhibit many interesting properties, especially advantages on unstructured meshes and solving steady state problems, see e.g. [38,39].…”

“…This issue makes it difficult to determine the convergence criterion for the iteration and challenging to apply the method to complex problems in real applications. Studies on high order WENO schemes on unequal-sized sub-stencils reveal that they improve the convergence of classical high order WENO schemes to steady state solutions so that the residue of TVD Runge-Kutta iterations settles down to machine zero / round off errors [39]. In this paper, to resolve that open issue in high order fixed-point fast sweeping methods, we apply the fifth order multi-resolution WENO scheme in [37] to the fifth order fixed-point sweeping scheme and obtain an absolutely convergent fixed-point fast sweeping method for steady state of hyperbolic conservation laws, namely, the residue of the fast sweeping iterations converges to machine zero / round off errors for all benchmark problems tested.…”