A high-order Newton multigrid method for steady-state shallow water equations

TL;DR

This paper introduces a high-order Newton multigrid method using WENO discretization and Jacobian approximation to efficiently simulate steady-state shallow water flows with complex geometries, achieving high accuracy and computational efficiency.

Contribution

It develops a novel Jacobian approximation strategy within a Newton multigrid framework for shallow water equations, reducing computational costs while maintaining convergence.

Findings

Achieves third-order accuracy in numerical experiments.

Significantly reduces computation time compared to full Jacobian methods.

Demonstrates robustness across various flow scenarios.

Abstract

A high-order Newton multigrid method is proposed for simulating steady-state shallow water flows in open channels with regular and irregular geometries. The method integrates two components: (1) a finite volume discretization with third-order weighted essentially non-oscillatory (WENO) reconstruction for the governing shallow water equations, (2) a Newton-multigrid method with an efficient approximation of the Jacobian matrix for the resulting discrete system. Generating the full Jacobian matrix in Newton iterations causes substantial computational costs. To address this problem, we observe that the majority of the non-zero elements in the matrix exhibit negligible magnitudes. By eliminating these elements, we approximate the Jacobian matrix with fewer stencils, thereby significantly reducing calculation time. Numerical results demonstrate that the proposed simplification strategy improves computational efficiency while maintaining convergence rates comparable to those of the full Jacobian approach. Furthermore, the geometric multigrid method with a successive over-relaxation fast-sweeping smoother is employed for the linearized system to optimize performance. A variety of numerical experiments, including one-dimensional smooth subcritical flow, flows over a hump, and two-dimensional hydraulic jump over a wedge, are carried out to illustrate the third-order accuracy, efficiency and robustness of the proposed method.