Abstract
We present a numerical analysis of a time discretization method applied to Richards' equation. Written in its saturation-based form, this nonlinear parabolic equation models water flow into unsaturated porous media. Depending on the soil parameters, the diffusion coefficient may vanish or explode, leading to degeneracy in the original parabolic equation. The numerical approach is based on an implicit Euler time discretization scheme and includes a regularization step, combined with the Kirchhoff transform. Convergence is shown by obtaining error estimates in terms of the time step and of the regularization parameter.
Similar content being viewed by others
References
H.W. Alt and S. Luckhaus, Quasilinear elliptic–parabolic differential equations, Math. Z. 183 (1983) 311–341.
R.G. Baca, J.N. Chung and D.J. Mulla, Mixed transform finite element method for solving the non-linear equation for flow in variably saturated porous media, Internat. J. Numer. Methods Fluids 24 (1997) 441–455.
P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuss, H. Rentz-Reichert and C. Wieners, UG – A flexible software toolbox for solving partial differential equations, Comput. Vis. Sci. 1 (1997) 27–40.
J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media(Kluwer Academic, Dordrecht, 1991).
L. Berganaschi and M. Putti, Mixed finite elements and Newton-type linearizations for the solution of Richards' equation, Internat. J. Numer. Methods Engrg. 45 (1999) 1025–1046.
M.A. Celia, E.T. Bouloutas and R.L. Zarba, A general mass-conservative numerical solution for the unsaturated flow equation, Water Resour. Res. 26 (1990) 1483–1496.
C.J. van Duijn and L.A. Peletier, Nonstationary filtration in partially saturated porous media, Arch. Rational Mech. Anal. 78 (1982) 173–198.
C.M. Elliott, Error analysis of the enthalpy method for the Stefan problem, IMA J. Numer. Anal. 7 (1987) 61–71.
R. Eymard, M. Gutnic and D. Hillhorst, The finite volume method for Richards equation, Comput. Geosci. 3 (1999) 259–294.
M.Th. van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsatu-rated soils, Soil Sci. Soc. Amer. J. 44 (1980) 892–898.
B.H. Gilding, Qualitative mathematical analysis of the Richards equation, Transp. Porous Media 5 (1991) 651–666.
W. Hackbusch, On first and second order box schemes, Computing 41 (1989) 277–296.
J. Hulshof and N. Wolanski, Monotone flows in N-dimensional partially saturated porous media: Lipschitz continuity of the interface, Arch. Rational Mech. Anal. 102 (1988) 287–305.
W. Jäger and J. Ka¡cur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes, Math. Model. Numer. Anal. 29 (1995) 605–627.
J. Ka¡ cur, Solution to strongly nonlinear parabolic problems by a linear approximation scheme, IMA J. Numer. Anal. 19 (1999) 119–145.
D. Kavetski, P. Binning and S.W. Sloan, Adaptive time stepping and error control in a mass conserva-tive numerical solution of the mixed form of Richards equation, Adv. Water Res. 24 (2001) 595–605.
C.T. Kelley, C.T. Miller and M.D. Tocci, Termination of Newton/chord iterations and the method of lines, SIAM J. Sci. Comput. 19 (1998) 280–290.
J.L. Lions and E. Magenes, Non Homogenous Boundary Value Problems and Applications,Vol.I (Springer, Berlin, 1972).
E. Magenes, R.H. Nochetto and C. Verdi, Energy error estimates for a linear scheme to approximate nonlinear parabolic problems, Math. Model. Numer. Anal. 21 (1987) 655–678.
Y. Mualem, A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res. 12 (1976) 513–522.
R.H. Nochetto and C. Verdi, Approximation of degenerate parabolic problems using numerical inte-gration, SIAM J. Numer. Anal. 25 (1988) 784–814.
F. Otto, L 1-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations 131 (1996) 20–38.
I.S. Pop, Regularization methods in the numerical analysis of some degenerate parabolic equations, Preprint 98-43 (SFB 359), IWR, University of Heidelberg, Germany (1998).
I.S. Pop and W.A. Yong, A numerical approach to degenerate parabolic equations, Numer. Math. (in press).
F. Radu, I.S. Pop and P. Knabner, Error estimates for an Euler implicit, mixed finite element dis-cretization of Richards' equation: Equivalence between mixed and conformal approaches, RANA Preprint 02-06, Eindhoven University of Technology (2002).
M.J.L. Robin, A.L. Gutjahr, E.A. Sudicky and J.L. Wilson, Cross-corelated random field generation with the direct Fourier transform method, Water Resour. Res. 29 (1993) 2385–2397.
M. Slodicka, Some finite element schemes arising in modeling of flow through porous media, Habil. dissertation, University of Augsburg, Germany (1999).
C. Wagner, G. Wittum, R. Fritsche and H.P. Haar, Diffusions–Reaktionsprobleme in ungesättigten porösen Medien, in: Mathematik–Schlüsseltechnologie für die Zukunft, eds. K.H. Hoffmann, W. Jäger, T. Lohmann and H. Schunck (Springer, Berlin, 1997) pp. 243–253.
C. Wagner, Numerical methods for diffusion-reaction-transport processes in unsaturated porous me-dia, Comput. Vis. Sci. 1 (1998) 97–105.
G.A. Williams and C.T. Miller, An evaluation of temporally adaptive transformation approaches for solving Richards' equation, Adv. Water Res. 22 (1999) 831–840.
C.S. Woodward and C.N. Dawson, Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media, SIAM J. Numer. Anal. 37 (2000) 701–724.
E. Zeidler, Applied Functional Analysis, Vol. I, Applied Mathematical Sciences, Vol. 108 (Springer, New York, 1995).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pop, I.S. Error Estimates for a Time Discretization Method for the Richards' Equation. Computational Geosciences 6, 141–160 (2002). https://doi.org/10.1023/A:1019936917350
Issue Date:
DOI: https://doi.org/10.1023/A:1019936917350