Application of Acceleration Technique to Improve the Convergence of Nonlinear Solvers for the Solution of Richards Equation Using the Finite Volume method

Document Type : Original Article

Authors

1 PhD student, Water Engineering Department., Ferdowsi University of Mashhad., Mashhad., Iran

2 Professor, Water Engineering Department, Ferdowsi University of Mashhad., Mashhad., Iran

3 Associate Professor, Water Engineering Department, Ferdowsi University of Mashhad., Mashhad., Iran

Abstract

Groundwater flow in variably saturated soils is described by the nonlinear Richards equation. The solution of Richards equation using implicit finite volume method produces a system of nonlinear equations, whose resolution demands for the application of an appropriate nonlinear systems solvers, such as the Picard or the Newton schemes. The Picard iterative technique is a robust method, which is convergent at a linear rate while the Newton-Raphson (Newton) method can produce accurate results when using a suitable initial guess and converge at a much higher rate. But since the computational cost of Newton method, due to the calculation of the formal inversion of Jacobian Matrix and partial derivatives is high, this method doesn’t seem to be demanding for the problems with more than one dimension. In this paper, to reduce the cost and time of Newton’s iterations, the mixed form of Picard and quasi-Newton followed by the Broyden method is employed. Therefore the purpose of this paper is first to investigate the effect of three linearization methods on the time of calculation and then to study the accuracy of the proposed implicit finite volume method for solving two dimensional mixed form of Richards equation. For this, data from a test case for a sandy clay loam soil with constant head as a boundary condition and initial moisture close to residual moisture was used. Comparison of the three linearized methods showed that for well selected convergence criterion, Picard/quasi Newton algorithm can dramatically reduce the time of calculations and number of iteration when compared to Picard scheme. Broyden method has a small effect on reducing time of calculation comparing with Picard/quasi Newton method. The performance of the proposed numerical method was then studied for two dimensionalmovement of water in a variably saturated soil using Picard/quasi Newton/Broyden method in comparison with the Warrick’s analytical solution. The results showed that the implicit finite volume method produced a good fit to the analytical solution, so that the RMSE=0.0001 was calculated between the models results and analytical solution on Richards equation.
 

Keywords

References

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Iranian Journal of Irrigation & Drainage
Volume 11, Issue 4 - Serial Number 64
November and December 2017
Pages 517-527

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