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A fully convergent fixed-point fast sweeping method with the WENO-JS local solver for steady state of hyperbolic conservation laws

Liang Li, Jun Zhu, Yong-Tao Zhang

arXiv:2609.38841v1math.NAcs.NA

TL;DR

Classical WENO-JS fast sweeping can struggle to drive steady-state iteration residues to round-off levels, while unequal-substencil alternatives are more complex and costly. The paper freezes WENO-JS nonlinear weights after residue stabilization, using a simple criterion to select the freezing step. Experiments report round-off residuals and full convergence, with greater efficiency than two comparison methods.

  • Problem

    With classical WENO-JS, residues can remain above round-off levels on difficult steady-state problems, while unequal-substencil alternatives are generally more complex and costly.

    Source: §1 · p. 2

  • Method

    The method freezes WENO-JS nonlinear weights after residue stabilization and uses a simple, robust approach to choose the freezing step.

    Source: §1 · pp. 2–3

  • Results

    The scheme consistently reaches round-off residuals and is more efficient than fully convergent MRWENO fast sweeping [15] and TVD-RK3 time marching.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The experiments show that the classical WENO-JS solver can achieve full convergence while retaining its essentially non-oscillatory property and order of accuracy.

    Source: §4 · p. 21

    Freezing is applied after residue stabilization, when discontinuities are resolved and essentially fixed and subsequent solution updates are negligible.

    Source: §2.3.2 · p. 9

Abstract

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The fixed-point fast sweeping methods with weighted essentially non-oscillatory (WENO) local solvers are a class of efficient and high-order accuracy numerical methods for solving steady-state solutions of hyperbolic conservation laws. However, with the classical WENO-JS local solver, the iteration residue of high-order fixed-point fast sweeping scheme often has difficulty to settle down to round-off errors. To achieve the full convergence in a fast sweeping method, the fixed-point fast sweeping methods with non-traditional WENO local solvers based on unequal-sized substencils were designed. However, the WENO schemes based on unequal-sized stencils are more complex and in general more expensive in computational costs than the classical WENO-JS schemes. In this paper, we go back to the classical WENO-JS local solver and develop a new fully convergent fifth-order fixed-point fast sweeping method for solving steady-state problems of hyperbolic conservation laws. Based on recent studies on the nonlinear weighting process around discontinuities of solution, we apply the technique of frozen weights for avoiding unnecessary adjustment of nonlinear weights in the WENO-JS local solver, which freezes the nonlinear weights once the residual sequence in the fast sweeping iterations has stabilized. Different from the existing work on frozen weights, we design a simple and robust approach to judge stabilization of iteration residues and determine the iteration step when the nonlinear weights are frozen in the fast sweeping method. Extensive numerical experiments on a wide range of challenging two-dimensional steady-state problems demonstrate that, unlike the previous fast sweeping method with the fifth-order WENO-JS local solver, the proposed new scheme consistently drives the iteration residues to round-off errors and achieves the full convergence.

1 Introduction

The introduction frames fully convergent steady-state computation as a challenge for fixed-point fast sweeping with the classical WENO-JS solver. The paper proposes freezing its nonlinear weights after residue stabilization and reports full convergence on challenging two-dimensional problems.

1 Introduction

  • WENO spatial discretizations provide high-order accuracy in smooth regions and essentially non-oscillatory transitions across discontinuities, while fixed-point fast sweeping offers explicit iterations for nonlinear steady-state systems.Fast sweeping uses alternating sweeps with Gauss-Seidel iterations.Source: §1 · pp. 1–2
  • Classical WENO-JS fast sweeping can fail to reduce residues to round-off levels on difficult problems, while unequal-substencil alternatives are generally more complex and costly.Residue errors may dominate simulation errors and complicate convergence criteria.Source: §1 · p. 2
  • The method returns to the classical WENO-JS solver and freezes its nonlinear weights once the fast-sweeping residue sequence stabilizes.The paper introduces a simple, robust approach for identifying stabilization and choosing when to freeze the weights.Source: §1 · pp. 2–3
  • The proposed scheme consistently reaches round-off residuals and is more efficient than fully convergent MRWENO fast sweeping [15] and TVD-RK3 time marching.It also improves on the original fifth-order WENO-JS fast sweeping method.Source: §1 · p. 3

2 Description of the numerical method

The method combines fifth-order WENO-JS spatial discretization with fixed-point fast sweeping, then freezes nonlinear weights after a residue-based stabilization test. The resulting procedure is designed to carry the iteration beyond the stagnation that can affect the classical solver.

2.1 The fifth-order WENO-JS reconstruction for ˆf +

  • The fifth-order WENO-JS reconstruction combines three third-order reconstructions built on overlapping three-cell substencils, with nonlinear weights based on smoothness indicators.The reconstruction is applied to fluxes after splitting them into positive and negative parts for upwinding.Source: §2 · p. 3, §2.1 · p. 4

2.2 The fixed-point fast sweeping method

  • The fixed-point fast sweeping solver applies Gauss-Seidel updates using the newest available stencil values and repeatedly sweeps in four alternating directions.The nonlinear function represents the fifth-order WENO-JS discretization and depends on 13 stencil values.Source: §2.2 · p. 5

2.3 The fully convergent fifth-order fast sweeping WENO-JS scheme

  • Freezing nonlinear weights after residue stabilization avoids unnecessary weight adjustments and lets the residue continue toward machine zero while maintaining solution accuracy and the essentially non-oscillatory property.The method stores the weights and reuses them as linear weights for subsequent flux computations.Source: §2.3 · p. 6, §2.3.2 · p. 8

2.3.1 The criterion for assessing the residue sequence

  • The stabilization test uses a relative-error criterion because conventional WENO-JS residues can settle at different levels across test cases.The selected parameters are m = 100 and ε = 10^-3; the reported tests identify stabilization across the numerical experiments.Source: §2.3.1 · p. 6

2.3.2 Freezing the Nonlinear Weights

  • When the residue stabilizes, the method stores the nonlinear weights and uses them directly as linear weights in later flux computations.Weights are stored separately for each of the four sweep directions.Source: §2.3.2 · pp. 8–9
  • The frozen-weight strategy is stated to extend beyond WENO-JS to variants such as WENO-Z when locally computed weights oscillate after the residue levels off.Preliminary tests with WENO-Z reportedly showed essentially identical convergence behavior and computational savings.Source: §2.3.2 · p. 9
  • The authors explain that freezing preserves ENO behavior because weights have already suppressed stencils crossing discontinuities when the solution is near steady state.They state that subsequent solution changes are negligible and the frozen combination continues to favor smooth substencils near shocks.Source: §2.3.2 · p. 9

3 Numerical experiments

The experiments compare the proposed fast-sweeping scheme with WENO-JS, multi-resolution WENO, and Runge–Kutta variants on accuracy and challenging steady-state problems. They also test a case where the original WENO-JS iteration converges, to check whether weight freezing triggers prematurely and adds excess cost.

Example 1. 2D accuracy test

  • All four schemes achieve their designed order in the two-dimensional accuracy test, and fast sweeping saves about 70% of CPU time versus Runge–Kutta.The weight-freezing check adds negligible cost in this case, where residuals decrease smoothly and neither AC scheme freezes weights.Source: §3 · p. 10, Table 2 · p. 10

Example 2. Regular shock reflection

  • For this problem, FS-WENOJS-AC converges at CFL 1 versus FS-MRWENO’s maximum 0.6, takes 15.42 seconds—reducing CPU time by more than half—and requires more iterations.FS-WENOJS fails to achieve full convergence even with a very small CFL number; the converged schemes’ density contours closely resemble one another.Source: Example 2. Regular shock reflection · p. 11

Example 3. Supersonic flow past one short plate with an attack angle

  • For the one-short-plate case, FS-WENOJS stagnates near 10^-3.3 even at CFL 0.1, while the other three schemes converge.RK-WENOJS-AC’s fastest run is twice as slow as FS-WENOJS-AC’s, and all three converged schemes produce similar pressure contours.Source: Example 3. Supersonic flow past one… · p. 12

Example 4. Supersonic flow past two short plates with an attack angle

  • With two short plates, FS-WENOJS again fails to reach machine zero; FS-WENOJS-AC uses about 60% less CPU time than FS-MRWENO and 30% less than Runge–Kutta.The three schemes’ pressure contours are nearly identical.Source: Example 4. Supersonic flow past two… · pp. 12, 14

Example 5. Supersonic flow past three long plates with an attack angle

Example 6. 30° steady oblique shock problem

Example 7. Supersonic flows past an airfoil

Example 8. Subsonic flows past an airfoil

4 Concluding remarks

The paper introduces a fully convergent fixed-point fast sweeping method using the classical fifth-order WENO-JS solver with equal-sized substencils. Its experiments report full convergence across the tested examples and approximately twice the computational speed of other fully converged methods.

  • The new method uses the classical fifth-order WENO-JS local solver with equal-sized substencils and a simple frozen-weight strategy.The strategy eliminates unnecessary nonlinear-weight adjustments near discontinuities.Source: §4 · p. 21
  • The scheme consistently achieves full convergence across the tested two-dimensional steady flow examples, unlike the original fifth-order WENO-JS fast sweeping scheme.The reported method preserves order of accuracy and the essentially non-oscillatory property while driving the iteration residual to machine zero.Source: §4 · p. 21
  • Approximately twice faster in computational costs to reach full convergence than FS-MRWENO and RK-WENOJS-AC.Source: §4 · p. 21
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