Enhanced Wu's Method

• Enhanced Wu's Method is an advanced solver combining fixed-point fast sweeping and fifth-order MR-WENO reconstruction to achieve convergence down to machine roundoff error.
• It employs alternating directional Gauss–Seidel sweeps and local stencil adaptivity to accelerate convergence and significantly enhance computational efficiency.
• Benchmark tests demonstrate a 40-60% reduction in CPU time while maintaining high-order accuracy and robust performance across steady-state hyperbolic conservation laws.

Enhanced Wu's Method refers to the advancement of fixed-point fast sweeping weighted essentially non-oscillatory (WENO) schemes for achieving absolutely convergent solutions to steady-state hyperbolic conservation laws. This methodology replaces conventional WENO flux reconstructions with a fifth-order multi-resolution WENO scheme, thereby eliminating the stagnation of iterative residue at truncation error and ensuring convergence to machine roundoff errors for all test cases. The resulting approach integrates high-order accuracy, explicit update structure, Gauss–Seidel acceleration, alternating directional sweeps, and local stencil adaptivity, substantially improving both convergence characteristics and computational efficiency (Li et al., 2020).

1. Fixed-Point Fast Sweeping WENO Schemes

The original fixed-point fast sweeping method of Wu et al. targets steady-state hyperbolic conservation laws of the form

$f(u)_x + g(u)_y = R(u, x, y)$

on Cartesian grids. The numerical strategy involves:

• Flux splitting $f(u) = f^+(u) + f^-(u)$, $g(u) = g^+(u) + g^-(u)$,
• Semi-discrete residuals formed using WENO numerical fluxes,
• Forward Euler time-marching for the fixed-point iteration:

$$u_{ij}^{n+1} = u_{ij}^n + \Delta t_n L_{ij}(\mathbf u^n)$$

• Adaptive time step sizing based on local maximum characteristic speeds and a chosen CFL number.

The convergence rate of naive Jacobi iterations is slow unless a small CFL is imposed. In contrast, the fast sweeping variant introduces:

• Immediate Gauss–Seidel update of each cell as soon as its new value is available,
• Directionally alternating sweeps in all four possible index orders, fostering rapid propagation of information along characteristics,
• The use of standard fifth-order Jiang–Shu WENO reconstruction employing smoothness indicators and WENO-Z weights for numerical fluxes.

2. Multi-Resolution WENO Reconstruction

The enhancement proposed by Zhu & Shu involves a fifth-order multi-resolution (MR) WENO reconstruction, in which:

• Fluxes at interfaces are reconstructed not from fixed-sized equally spaced stencils but from a hierarchy of three nested, unequal sub-stencils.
• For each interface, interpolating polynomials of degree 2, 4, and 6 are built on progressively wider stencils centered at the interface.
• These polynomials are combined via nonlinear weights computed from smoothness indicators using a partition-of-unity design, with explicit formulas for sub-polynomial weights and smoothness indicators.
• The MR-WENO method ensures local adaptivity of stencils and more robust suppression of spurious oscillations.

The general reconstruction at an interface utilizes: $$\hat h_{i+\frac12} = \sum_{k=1}^3 \omega_k\, p_k(x_{i+\frac12})$$ where $p_k(x)$ are degree-4 “sub-polynomials” derived by carefully forming linear combinations of the base interpolating polynomials, and the $\omega_k$ are nonlinear weights favoring smooth local stencils.

3. Integration Into the Fast Sweeping Framework

The Enhanced Wu’s Method replaces the standard WENO reconstructions inside the fixed-point fast sweeping algorithm with the aforementioned MR-WENO procedure. The steps, denoted as Algorithm FE–MR–Fast–Sweeping, are:

1. Initialize the state using boundary data or a guess.
2. Precompute characteristic speeds and set the CFL parameter.
3. Iteratively:
   • Update the time step according to local speeds.
   • Perform four alternating directional Gauss–Seidel sweeps, reconstructing all fluxes via the MR-WENO approach using most up-to-date stencil points.
   • Explicitly update each cell’s value and immediately overwrite for subsequent stencils.
4. After each total sweep, check convergence using the average residual

$$\mathrm{ResA} = \frac1{NM} \sum_{i,j} \left| u_{ij}^{n+1} - u_{ij}^n \right| / \Delta t_n$$

and halt when it is sufficiently small (e.g., $< 10^{-12}$).

Each flux is re-reconstructed locally whenever a neighbor is updated, leveraging full stencil adaptivity without global rebuilding requirements.

4. Absolute Convergence Properties

The defining feature of the enhanced method is “absolute convergence”: as iteration proceeds,

$$\lim_{n\to\infty} \mathrm{ResA}^n = \mathcal{O}(\text{machine roundoff}) \ll 10^{-12}$$

with no plateauing at higher truncation errors, unlike previous schemes. The residual is monitored not via the explicit local residual—identically zero in the method—but via the normalized change between iterations, which in practice tracks the actual error decay.

Under standard Lipschitz-continuity assumptions on the flux Jacobians, it can be established that for CFL numbers below a stability threshold,

$$\gamma < \gamma_{\max} = \frac{\Delta x\,\Delta y}{\Delta x\,\alpha_y+\Delta y\,\alpha_x},$$

the fixed-point map is a strict contraction in the $\ell^\infty$ norm. Crucially, the four alternating Gauss–Seidel sweeps amplify contractivity, resulting in geometric decay of the residual to machine zero. Taylor expansion arguments indicate each MR-WENO reconstruction's Lipschitz constant is close to unity, preserving stability and acceleration.

5. Benchmark Results and Effectiveness

Standard one- and two-dimensional steady-state test problems demonstrate:

• Fifth-order $L^1$ and $L^\infty$ accuracy for smooth solutions,
• Fast decay of the average residual to $10^{-16}$ (machine zero) with no stagnation,
• CPU time reduced by $40\%$–$60\%$ compared to TVD–RK3 Jacobi methods.

A representative performance comparison for the regular shock-reflection problem on a $120\times30$ mesh is summarized as:

Method	CFL $\gamma$	Iterations	Final ResA	CPU Time (s)
FE Jacobi	0.1	12,046	$10^{-12}$	618.4
RK3 Jacobi	0.6	5,634	$10^{-12}$	289.9
FE–MR–FS (enhanced)	0.6	1,934	$10^{-13}$	98.9

Residual plots exhibit geometric straight-line decay of $\log_{10}(\mathrm{ResA}^n)$ down to machine precision. Comparable explicit schemes using Jacobi or RK-WENO iterations stagnate at $10^{-9}$–$10^{-11}$.

6. Significance and Broader Impact

The Enhanced Wu's Method provides an explicit, high-order, absolutely convergent solver for steady-state hyperbolic conservation laws. It combines methodological advances in multi-resolution WENO reconstruction and fast sweeping acceleration, delivering robust, efficient, and scalable convergence across all benchmark scenarios (Li et al., 2020). The approach is broadly applicable to general hyperbolic equations, accommodates monotone numerical fluxes, and efficiently leverages high-order approximations. This capability addresses a longstanding challenge of residue stagnation in high-order iterative schemes, thus facilitating the routine application of high-accuracy explicit solvers in computational fluid dynamics and related domains.