Consistent Nonlinear Solver for Solute Transport in Variably Saturated Porous Media

We propose a new Jacobian-free solver for the system of nonlinear equations describing transport of a nonreactive solute in porous media. Maximum principles (MPs) are important properties of the solution and impose severe requirements on the discretization and nonlinear solver. Exact local water balance is needed to show the MP for the solute concentration. The proposed solver guarantees the discrete MPs for the solute with the tolerance of linear solvers (typically \(10^{-10}\)) even when the tolerance of the nonlinear solver is not tight (typically \(10^{-5}\)). The proposed technique combines the stable discretization of the Frechet derivative of the continuous functional describing the system with the slope-limiting algorithm for water retention models.