  This study concerns numerical methods for efficiently solving the Richards
equation where different weak formulations and computational techniques are
analyzed. The spatial discretizations are based on standard or mixed finite
element methods. Different implicit and semi-implicit temporal discretization
techniques of second-order accuracy are studied. To obtain a linear system for
the semi-implicit schemes, we propose second-order techniques using
extrapolation formulas and/or semi-implicit Taylor approximations for the
temporal discretization of nonlinear terms. A numerical convergence study and a
series of numerical tests are performed to analyze efficiency and robustness of
the different schemes. The developed scheme, based on the proposed temporal
extrapolation techniques and the mixed formulation involving the saturation and
pressure head and using the standard linear Lagrange element, performs better
than other schemes based on the saturation and the flux and using the
Raviart-Thomas elements. The proposed semi-implicit scheme is a good
alternative when implicit schemes meet convergence issues.
