Ross scheme, Newton-Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation

The solution of the mathematical model for flow in variably saturated porous media described by the Richards equation (RE) is subject to heavy numerical difficulties due to its highly nonlinear properties and remains very challenging. Two different algorithms are used in this work to solve the mixed form of RE: the traditional iterative algorithm and a time-adaptive algorithm consisting of changing the time-step magnitude within the iteration procedure while the nonlinear parameters are computed with the state variable at the previous time. The Ross method is an example of this type of scheme, and we show that it is equivalent to the Newton-Raphson method with a time-adaptive algorithm. Both algorithms are coupled to different time-stepping strategies: the standard heuristic approach based on the number of iterations and two strategies based on the time truncation error or on the change in water saturation. Three different test cases are used to evaluate the efficiency of these algorithms. The numerical results highlight the necessity of implementing an estimate of the time truncation errors.