A Newton multigrid method for steady-state shallow water equations with topography and dry areas

TL;DR

This paper introduces a Newton multigrid method tailored for steady-state shallow water equations that effectively handles topography and dry areas, demonstrating high efficiency and robustness through numerical experiments.

Contribution

It presents a novel Newton multigrid approach with residual-based Jacobian regularization for steady-state SWEs with wet/dry transitions, improving solution robustness.

Findings

Efficient convergence demonstrated in numerical tests.

Method effectively handles wet/dry transitions.

Maintains well-balanced property across scenarios.

Abstract

The paper develops a Newton multigrid (MG) method for one- and two-dimensional steady-state shallow water equations (SWEs) with topography and dry areas.It solves the nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs by using Newton's method as the outer iteration and a geometric MG method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton MG method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady-state problem with wet/dry transitions. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton MG method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.