A high-order Newton multigrid method with a simplified Jacobian for steady-state shallow water equations
For computational fluid dynamics researchers, this method offers a more efficient way to solve steady-state shallow water equations with high-order accuracy, though it is an incremental improvement over existing Newton multigrid techniques.
The paper proposes a high-order Newton multigrid method for steady-state shallow water equations that uses a simplified Jacobian approximation to reduce computational cost while maintaining convergence behavior comparable to the full Jacobian approach. Numerical experiments demonstrate third-order accuracy and significant efficiency improvements.
A high-order Newton multigrid method is proposed for steady-state shallow water flows in open channels with regular and irregular geometries. The method integrates a finite volume discretization with third-order weighted essentially non-oscillatory (WENO) reconstruction and a Newton multigrid framework with an efficient approximation of the Jacobian matrix for solving the resulting discrete system. In high-order schemes, the computational cost of Jacobian construction becomes dominant due to the wide stencil. Meanwhile, only a small fraction of the non-zero Jacobian entries exhibit large magnitudes. Based on this observation, a simplified Jacobian approximation is introduced using reduced stencils, in which selected off-stencil contributions are neglected, thereby achieving a substantial reduction in computational cost. The proposed approach is verified numerically to show significant efficiency improvement while maintaining comparable convergence behavior to that obtained with the full Jacobian approach. To further enhance performance, a geometric multigrid method incorporating a successive over-relaxation iteration as the smoother is applied to solve the linear systems arising in each Newton step. A variety of numerical experiments, including a one-dimensional smooth subcritical flow, flows over a hump, and a two-dimensional hydraulic jump over a wedge, are carried out to illustrate the third-order accuracy, efficiency, and robustness of the proposed method.