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.

这项研究涉及数值方法，可以有效地解决Richards方程，其中分析了不同的弱公式和计算技术。空间离散化基于标准或混合有限元方法。研究了二阶精度的不同隐式和半隐式时间离散技术。为了获得半隐式方案的线性系统，我们提出了使用外推公式和/或半隐式泰勒逼近法对非线性项进行时间离散化的二阶技术。进行了数值收敛研究和一系列数值测试，以分析不同方案的效率和鲁棒性。制定的方案 基于提出的时间外推技术以及涉及饱和度和压头的混合公式，并使用标准线性Lagrange元素，与基于饱和度和通量并使用Raviart-Thomas元素的其他方案相比，其性能更好。当隐式方案遇到收敛问题时，建议的半隐式方案是一个很好的选择。