An Efficient Solver for the Richards Equation for Variably Saturated Flows in Porous Media

We present a nonlinear multigrid solver for the Richards equation in variably saturated porous media with strongly nonlinear hydraulic conductivity and water-retention relationships. The governing equation is discretized using a second-order conservative finite-difference scheme in space and an implicit backward differentiation formula in time. The core component of the solver is a nonlinear Gauss--Seidel (NGS) smoother based on a triangular splitting of the diffusion operator combined with diagonal stabilization. This construction yields a sequence of locally decoupled scalar nonlinear problems that can be solved efficiently and robustly using only a few Newton iterations. Under suitable monotonicity assumptions, we establish the convergence of the NGS iteration in the $L^\infty$ norm and derive explicit conditions on the stabilization parameters. Numerical experiments for benchmark infiltration, drainage, and root-uptake problems demonstrate that the proposed NGS-based multigrid framework is both computationally efficient and robust.