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

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.