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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

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2191-0294
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Volume 19, Issue 3-4

Issues

RBFPUM with QR Factorization for Solving Water Flow Problem in Multilayered Soil

Mohamed Sadik / El Hassan Ben-Ahmed / Mohamed Wakrim
Published Online: 2018-04-05 | DOI: https://doi.org/10.1515/ijnsns-2017-0162

Abstract

We discuss the numerical modeling of infiltration in variably-saturated porous media. Richards’ equation is used to describe the infiltration towards the water table. It is difficult to accurately approximate its solution especially when we deal with layered soil due to its highly nonlinear fact. In this work, the nonlinearity is handled by using Gardner model. In the case of homogeneous soil, the linearized equation is solved using radial basis function partition of unity method (RBFPUM) with the introduction of QR factorization of Gaussian in order to enhance the numerical solution for small values of the so-called “shape parameter.” In the case of layered soil, domain decomposition principle is introduced. It is based on decomposing the general problem into many subproblems. The latter are solved by RBFPUM-QR and patched by using the Steklov–Poincaré equation. Infiltration towards water table in homogeneous and layered soil is considered as a numerical test.

Keywords: radial basis function; partition of unity; RBF-QR; Richards’ equation; infiltration; domain decomposition

PACS: 02.70.Jn

References

  • [1]

    Richards L. A., Capillary conduction liquids through porous mediums, J. Appl. Phys. 1 (5) (1931), 318–333.Google Scholar

  • [2]

    Celia M. A. and Bouloutas E. T., A general mass-conservative numerical solution for the unsaturated flow equation, Water Resour. Res. 26 (7) (1990), 1483–1496.CrossrefGoogle Scholar

  • [3]

    Gardner W., Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, Soil Sci. 85 (3) (1958), 228–232.CrossrefGoogle Scholar

  • [4]

    Stevens D. and Power H., A scalable and implicit meshless RBF method for the 3D unsteady nonlinear Richards equation with single and multi-zone domain, Int. J. Numer. Meth. Engng. 85 (2) (2011), 135–163.CrossrefGoogle Scholar

  • [5]

    Stevens D., Power H. and Morvan H., An order-N complexity meshless algorithm for transport-type PDEs, based on local Hermitian interpolation, Eng. Anal. Bound. Elem. 33 (4) (2009), 425–441.CrossrefWeb of ScienceGoogle Scholar

  • [6]

    Tracy F. T., Clean two– and three–dimensional analytical solutions of Richards’s equation for testing numerical solvers, Water Resour. Res. 42 (2006), W08503.Google Scholar

  • [7]

    Srivastava R. and Jim Yeh T. C., Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils, Resour Water. Res. 27 (5) (1991), 753–762.CrossrefGoogle Scholar

  • [8]

    Tracy F. T., 1–D, 2–D, and 3–D analytical solutions of unsaturated flow in groundwater, Hydrol J.. 170 (1–4) (1995), 199–214.CrossrefGoogle Scholar

  • [9]

    Warrick A. W., An analytical solution to Richards’ equation for time-varying infiltration, Water Resour. Res. 27 (5) (1991), 763–766.CrossrefGoogle Scholar

  • [10]

    Yuan F. and Lu Z., Analytical solutions for vertical flow in unsaturated, rooted soils with variable surface, Vadose Zone J. 4 (2005), 1210–1218.CrossrefGoogle Scholar

  • [11]

    Cockett R., Simulation of unsaturated flow using Richards equation, 2014. Available from: http://www.row1.ca/s/pdfs/courses/RichardsEquation.pdf, Accessed: 11 January 2017.Google Scholar

  • [12]

    Havercamp R., M. Vauclin, J. Touma, P. Wierenga J. and Vachaud G., A comparison of numerical simulation models for one–dimensional infiltration, Soil Sci. Soc. Am. J. 41 (1977), 285–294.Crossref

  • [13]

    Kirkland M. R. and Hills R. G., Algorithms for solving Richards’ equation for variably saturated soils, Water Resour. Res. 28 (8) (1992), 2049–2058.CrossrefGoogle Scholar

  • [14]

    van Dam J. C. and Feddes R. A., Numerical simulation of infiltration, evaporation and shallow groundwater levels with the Richards equation, J. Hydrol. 233 (1–4) (2000), 72–85.CrossrefGoogle Scholar

  • [15]

    Eymard R., Gutnic M. and Hilhorst D., The finite volume method for Richards equation, Comput. Geosci. 3 (3) (1999), 259–294.CrossrefGoogle Scholar

  • [16]

    Romano N., Brunone B. and Santini A., Numerical analysis of one–dimensional unsaturated flow in layered soils, Adv. Resour Water. 21 (4) (1998), 315–324.CrossrefGoogle Scholar

  • [17]

    Brunone B., Ferrante M., Romano N. and Santini A., Numerical simulations of one-dimensional infiltration into layered soils with the Richards equation using different estimates of the interlayer conductivity, Vadoze Zone J. 2 (2003), 193–200.CrossrefGoogle Scholar

  • [18]

    Kansa E. J., Multiquadrics–A scattered data approximation scheme with applications to computational fluid-dynamics–I surface approximations and partial derivative estimates, Comput. Math. Applic. 19 (8/9) (1990), 127–145.CrossrefGoogle Scholar

  • [19]

    Kansa E. J., Multiquadrics–A scattered data approximation scheme with applications to computational fluid dynamics–II Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Applic. 19 (8/9) (1990), 147–161.CrossrefGoogle Scholar

  • [20]

    Wendland H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1) (1995), 389–396.CrossrefGoogle Scholar

  • [21]

    Safdari–Vaighani A., Heryudono A. and Larsson E., A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications, Sci J.. Comput. 64 (2) (2015), 341–367.Web of ScienceCrossrefGoogle Scholar

  • [22]

    Fornberg B. and Wright G., Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl. 48 (2004), 853–867.CrossrefGoogle Scholar

  • [23]

    Forenberg B., Larsson E. and Flyer N., Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput. 33 (2) (2011), 869–892.CrossrefWeb of ScienceGoogle Scholar

  • [24]

    Fasshauer G. E. and McCourt M. J., Stable evaluation of Gaussian RBF interpolants, SIAM J. Scient. Comput. 34 (2) (2012), A737–A762.Google Scholar

  • [25]

    Fasshauer G. E. and McCourt M. J., Kernel-based Approximation Methods Using Matlab, World Scientific, Singapore, 2015.Google Scholar

  • [26]

    Cavoretto R., De Marchi S., De Rossi A., Perracchione E. and Santin G., Partition of unity interpolation using stable kernel-based techniques, Appl. Numer. Math. 116 (2017), 95–107.Web of ScienceCrossrefGoogle Scholar

  • [27]

    Quarteroni A. and Valli A., Domain Decomposition Method for Partial Differential Equation, Oxford University Press, England, 1999.Google Scholar

  • [28]

    Duan Y., Tang P. F., Huang T. Z. and Lai S. J., Coupling domain decomposition method in electrostatic problems, Comput. Phys. Commun. 180 (2009), 209–214.CrossrefWeb of ScienceGoogle Scholar

  • [29]

    Ling L. and Kansa E. J., Preconditioning for radial basis functions with domain decomposition methods, Math. Comput. Model. 40 (13) (2004), 1413–1427.CrossrefGoogle Scholar

  • [30]

    Sarra S. A. and Kansa E. J., Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations, Marshall University, California, 2009.Google Scholar

  • [31]

    Schaback R., Multivariate interpolation and approximation by translates of a basis function, in: Approximation Theory VIII 1, Approximation and Interpolation, Chui C. K.( and Schumaker L. L., Eds.), pp. 491–514, World Scientific, Singapore, 1995.Google Scholar

  • [32]

    Wendland H., Approximation Scattered Data, Press Cambridge University, Cambridge, 2005.Google Scholar

  • [33]

    Babuška I. and Melenk J. M., The partition of unity method, Int. J. Numer. Meth. Engng. 40 (1997), 727–758.CrossrefGoogle Scholar

  • [34]

    Shcherbakov V. and Larsson E., Radial basis function partition of unity methods for pricing vanilla basket options, Comput. Math. Appl. 71 (1) (2016), 185–200.Web of ScienceCrossrefGoogle Scholar

  • [35]

    Shepard D., A two–dimensional interpolation function for irregularly-spaced data. in: Proceedings of the 23rd ACM National Conference, pp. 517–524, 1968.Google Scholar

About the article

Received: 2017-07-30

Accepted: 2018-03-19

Published Online: 2018-04-05

Published in Print: 2018-06-26


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 3-4, Pages 397–407, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0162.

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