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Richards' equation is frequently used to model flow in unsaturated porous media. This model captures physical effects, such as sharp fronts in fluid pressures and saturations, which is considered for practical application to an extensive variety of hydrologic evaluations, at a range of scales of simulation. The numerical solution of Richards' equation is difficult not only because of these physical effects but also because of the mathematical problems that occur in dealing with the nonlinearities. The mixed form of Richards' equation with a finite difference discretization leads to a nonlinear numerical model. In this work a nested Newton-type algorithm is briefly derived and it is suggested that nested Newton-type for finite difference methods can be effectively implemented as well as comparable with the results of nested Newton-type algorithm for finite volume and used in numerical models of Richards' equation. Two test problems are solved with a judicious choice of the initial guess and the quadratic convergence rate is obtained for any time step size for all flow systems.

An application of nested Newton-type algorithm for finite difference method solving Richards' equation