Numerical solutions of nonlinear diffusion models

The Richards' equation is a nonlinear, parabolic, partial derivative equation describing the flow of water in an unsaturated porous medium. Analytical solutions of Richards' equation exist only for simplified cases, so most practical situations require a numerical solution. After a suitable discretization of the spatial and temporal domain, the finite difference method is applied. In so doing, a nonlinear system of equations is obtained, solved combining explicit Euler method and Crank-Nicolson method. However, such iterative method requires an high computational effort and dos not converge in some particular cases. The aim was to replace such method using a direct method. In this work the Frontal method, the Multifrontal method and the Strongly Implicit procedure are considered.