Groundwater flow in variably saturated soils is described by the nonlinear Richards’ equation. The massconservative finite volume discretization recently proposed in (Adv. Water Resour. 2004; 27:1199–1215) produces a nonlinear algebraic problem, whose resolution demands for the application of an appropriate iterative strategy, such as the Picard or the Newton scheme. The Picard iterative technique results in a robust fixed-point method, which is globally convergent at a linear or sub-linear rate. On the other hand, the Newton iterative technique shows better convergence rates, but the computational cost of the full calculation is still comparable to the one of the Picard schemes, due to the additional calculation of the Jacobian matrix and its formal inversion. In this work, we investigate two acceleration techniques to reduce the cost of Newton iterations. These techniques are respectively based on a linearization of the nonlinear Jacobian matrix in accordance with the quasi-Newton approach and on a Broyden-type rank-one correction for a fixed number of sub-iterations. Numerical experiments on a set of two-dimensional test case modeling the uptake of plant roots in a highly heterogeneous soil show the performance and the effectiveness of both acceleration schemes in comparison with the original Picard and Newton methods.

Acceleration techniques for the iterative resolution of the Richards equation by the finite volume method

BEVILACQUA, IVAN;CANONE, Davide;FERRARIS, Stefano
2009

Abstract

Groundwater flow in variably saturated soils is described by the nonlinear Richards’ equation. The massconservative finite volume discretization recently proposed in (Adv. Water Resour. 2004; 27:1199–1215) produces a nonlinear algebraic problem, whose resolution demands for the application of an appropriate iterative strategy, such as the Picard or the Newton scheme. The Picard iterative technique results in a robust fixed-point method, which is globally convergent at a linear or sub-linear rate. On the other hand, the Newton iterative technique shows better convergence rates, but the computational cost of the full calculation is still comparable to the one of the Picard schemes, due to the additional calculation of the Jacobian matrix and its formal inversion. In this work, we investigate two acceleration techniques to reduce the cost of Newton iterations. These techniques are respectively based on a linearization of the nonlinear Jacobian matrix in accordance with the quasi-Newton approach and on a Broyden-type rank-one correction for a fixed number of sub-iterations. Numerical experiments on a set of two-dimensional test case modeling the uptake of plant roots in a highly heterogeneous soil show the performance and the effectiveness of both acceleration schemes in comparison with the original Picard and Newton methods.
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finite volume; unstructured grids; nonlinear solver
I. Bevilacqua; D. Canone; S. Ferraris
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/78314
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