Journal cover Journal topic
Hydrology and Earth System Sciences An interactive open-access journal of the European Geosciences Union
doi:10.5194/hess-2016-622
© Author(s) 2016. This work is distributed
under the Creative Commons Attribution 3.0 License.
Research article
20 Dec 2016
Review status
This discussion paper is under review for the journal Hydrology and Earth System Sciences (HESS).
Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed-form of Richards' equation
Fadji Hassane Maina1,2 and Philippe Ackerer1 1Laboratoire d'Hydrologie et de Géochimie de Strasbourg, Univ. Strasbourg/EOST – CNRS, 1 rue Blessig, 67084 Strasbourg, France
2CEA-Laboratoire de Modélisation des Transferts dans l'Environnement, Bât. 225, F-13108 Saint Paul lez Durance cedex, France
Abstract. The solution of the mathematical model for flow in variably saturated porous media described by Richards equation (RE) is subject to heavy numerical difficulties due to its highly non-linear 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 state variable is kept constant. 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 of water saturation. Three different test cases are used to evaluate the efficiency of these algorithms. The numerical results highlight the necessity of implementing two types of errors: the iterative convergence error (maximum difference of the state variable between two iterations) and an estimate of the time truncation errors. The algorithms using these two types of errors together were found to be the most efficient when highly accurate results are required.


Citation: Hassane Maina, F. and Ackerer, P.: Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed-form of Richards' equation, Hydrol. Earth Syst. Sci. Discuss., doi:10.5194/hess-2016-622, in review, 2016.
Fadji Hassane Maina and Philippe Ackerer
Fadji Hassane Maina and Philippe Ackerer
Fadji Hassane Maina and Philippe Ackerer

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Short summary
In many fields like climate change, hydrology and agronomy, water movement in unsaturated soils is usually simulated using the Richards equation. However, this equation requires lot of computational efforts to be solved due to its highly non linear behavior which hampers its use in simulations. In this paper, we analyze and developed some numerical strategies and we evaluate their reliability and efficiency.
In many fields like climate change, hydrology and agronomy, water movement in unsaturated soils...
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