In this study, second-order temporal discretizations are analyzed for solving the coupled system of infiltration and solute transport in unsaturated porous media. The Richards equation is used to describe unsaturated flow, while the advection-dispersion equation (ADE) is used for modeling solute transport. The standard finite element discretization in space is utilized and four time-stepping methods are studied. Three of these methods require an iterative resolution to solve the Richards equation in its mixed form. In the remaining method, a novel technique is proposed to linearize the system of equations in time, and the iterative processes are avoided. In this method, a free stabilized parameter is introduced. Numerical tests are conducted to analyze the accuracy and efficiency of methods. The developed linear scheme based on the optimal free parameter is accurate and performs better in terms of efficiency since it offers a considerable gain in computational time compared to the other methods. The reliability and effectiveness of the developed semi-implicit scheme are investigated using numerical experiments for modeling water flow and solute transport in unsaturated soils.
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