Solving the nonlinear and nonstationary Richards equation with two-level adaptive domain decomposition (dd-adaptivity)

Kuraz, Michal; Mayer, Petr; Pech, Pavel

doi:10.1016/j.amc.2015.03.130

Cited by 9 publications

(3 citation statements)

References 24 publications

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“…However, it is difficult to obtain analytical solutions [6][7][8] because the high nonlinearity of the RE is related to the complex hydraulic properties of unsaturated soils [9,10]. Hence, numerical solutions of RE have been extensively studied to investigate groundwater flow [11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Application of an improved P(m)-SOR iteration method for flow in partially saturated soils

Zhu

Huang

2021

Comput Geosci

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This paper studies the potential of using the successive over-relaxation iteration method with polynomial preconditioner (P(m)-SOR) to solve variably saturated flow problems described by the linearized Richards' equation. The finite difference method is employed to numerically discretize and produce a system of linear equations. Generally, the traditional Picard method needs to re-evaluate the iterative matrix in each iteration, so it is time-consuming. And under unfavorable conditions such as infiltration into extremely dry soil, the Picard method suffers from numerical non-convergence. For linear iterative methods, the traditional Gauss-Seidel iteration method (GS) has a slow convergence rate, and it is difficult to determine the optimum value of the relaxation factor w in the successive over-relaxation iteration method (SOR). Thus, the approximate optimum value of w is obtained based on the minimum spectral radius of the iterative matrix, and the P(m)-SOR method is extended to model underground water flow in unsaturated soils. The improved method is verified using three test examples. Compared with conventional Picard iteration, GS and SOR methods, numerical results demonstrate that the P(m)-SOR has faster convergence rate, less computation cost, and good error stability. Besides, the results reveal that the convergence rate of the P(m)-SOR method is positively correlated with the parameter m. This method can serve as a reference for numerical simulation of unsaturated flow.

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Section: Introductionmentioning

confidence: 99%

Application of an improved P(m)-SOR iteration method for flow in partially saturated soils

Zhu

Huang

2021

Comput Geosci

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“…First attempts of solving the Richards equation numerically date back to [11] in the 1970s. Various methods for the numerical solution of the Richards equation have been studied and improved in the last decades [12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

NUMERICAL SIMULATION OF A SINGLE RING INFILTRATION EXPERIMENT WITH hp-ADAPTIVE SPACE-TIME DISCONTINUOUS GALERKIN METHOD

2021

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We present a novel hp-adaptive space-time discontinuous Galerkin (hp-STDG) method for the numerical solution of the nonstationary Richards equation equipped with Dirichlet, Neumann and seepage face boundary conditions. The hp-STDG method presented in this paper is a generalization of a hp-STDG method which was developed for time dependent non-linear convective-diffusive problems. We describe the method and the single ring experiment, and then we present a numerical experiment which clearly demonstrates the superiority of the hp-STDG method over a discontinuous Galerkin method based on a static fine mesh.

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“…Additionally, other advanced numerical methods for solving variable saturated flow problems exhibit high levels of accuracy, computational efficiency, or ease of implementation under certain conditions. These include the two-level adaptive domain decomposition (Kuraz et al, 2015), finite analytic method (Zhang et al, 2016), and Chebyshev's spectral method (Wu et al, 2020b). A system of linear equations derived from the RE need to be solved iteratively, such as by using the Picard iteration (PI) scheme.…”

Section: Introductionmentioning

confidence: 99%

Modelling Unsaturated Flow in Porous Media Using an Improved Picard Iteration Scheme

Zhu

2021

Preprint

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The numerical solution of various systems of linear equations describing fluid infiltration uses the Picard iteration (PI). However, because many such systems are ill-conditioned, the solution process often has a poor convergence rate, making it very time-consuming. In this study, a control volume method based on non-uniform nodes is used to discretize the Richards equation, and adaptive relaxation is combined with a multistep preconditioner to improve the convergence rate of PI. The resulting adaptive relaxed PI with multistep preconditioner (MP(m)-ARPI) is used to simulate unsaturated flow in porous media. Three examples are used to verify the proposed schemes. The results show that MP(m)-ARPI can effectively reduce the condition number of the coefficient matrix for the system of linear equations. Compared with conventional PI, MP(m)-ARPI achieves faster convergence, higher computational efficiency, and enhanced robustness. These results demonstrate that improved scheme is an excellent prospect for simulating unsaturated flow in porous media.

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Solving the nonlinear and nonstationary Richards equation with two-level adaptive domain decomposition (dd-adaptivity)

Cited by 9 publications

References 24 publications

Application of an improved P(m)-SOR iteration method for flow in partially saturated soils

Application of an improved P(m)-SOR iteration method for flow in partially saturated soils

NUMERICAL SIMULATION OF A SINGLE RING INFILTRATION EXPERIMENT WITH hp-ADAPTIVE SPACE-TIME DISCONTINUOUS GALERKIN METHOD

Modelling Unsaturated Flow in Porous Media Using an Improved Picard Iteration Scheme

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