A numerical scheme for solving Richards' equation based on convex optimization
The vertical movement of soil moisture is overdamped and energy driven. A numerical scheme is suggested in which the water content at the next time step is obtained as a solution of a certain optimization problem. The functional being minimized is the total energy (determines the general direction of the water content change) plus a term containing hydraulic conductivity (controls the magnitude of the change). The motivation for the scheme was to handle saturated flows, with the energy of oversaturated flow due to incompressibility of water being set to infinity. The scheme is unconditionally stable (energy is non-increasing) even with explicit in time treatment of hydraulic conductivities, in which case the arising optimization problem is convex.