  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.
