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.

翻译：这项研究涉及有效解决理查斯等式的数字方法,其中分析了不同的微弱配方和计算技术;空间分解以标准或混合限定要素方法为基础;研究了二阶精确度的不同隐含和半隐含时间分解技术;为获得半隐含计划线性系统,我们提议使用外推公式和/或半隐含泰勒近似值来有效解决非线性术语的暂时分解;进行了数字趋同研究和一系列数字测试,以分析不同计划的效率和稳健性;根据拟议的时间外推法和涉及饱和和压力头并使用标准线性拉格朗元素的混合配方法,采用基于饱和和通量性并使用Raviart-Thomas元素的其他方案,比其他方案效果更好;拟议的半隐含计划在满足趋同问题时是一种良好的替代方案。