New preconditioning strategy for Jacobian-free solvers for variably saturated flows with Richards’ equation

Lipnikov, Konstantin; Moulton, David; Svyatskiy, Daniil

doi:10.1016/j.advwatres.2016.04.016

New preconditioning strategy for Jacobian-free solvers for variably saturated flows with Richards’ equation

Journal Article · Fri Apr 29 00:00:00 EDT 2016 · Advances in Water Resources

DOI:https://doi.org/10.1016/j.advwatres.2016.04.016· OSTI ID:1256101

Lipnikov, Konstantin ^[1]; Moulton, David ^[1]; ^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

We develop a new approach for solving the nonlinear Richards’ equation arising in variably saturated flow modeling. The growing complexity of geometric models for simulation of subsurface flows leads to the necessity of using unstructured meshes and advanced discretization methods. Typically, a numerical solution is obtained by first discretizing PDEs and then solving the resulting system of nonlinear discrete equations with a Newton-Raphson-type method. Efficiency and robustness of the existing solvers rely on many factors, including an empiric quality control of intermediate iterates, complexity of the employed discretization method and a customized preconditioner. We propose and analyze a new preconditioning strategy that is based on a stable discretization of the continuum Jacobian. We will show with numerical experiments for challenging problems in subsurface hydrology that this new preconditioner improves convergence of the existing Jacobian-free solvers 3-20 times. Furthermore, we show that the Picard method with this preconditioner becomes a more efficient nonlinear solver than a few widely used Jacobian-free solvers.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC52-06NA25396

OSTI ID:: 1256101

Alternate ID(s):: OSTI ID: 1325352

Report Number(s):: LA-UR--15-27929

Journal Information:: Advances in Water Resources, Journal Name: Advances in Water Resources Journal Issue: C Vol. 94; ISSN 0309-1708

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (30)

Mixed finite elements and Newton-type linearizations for the solution of Richards' equation Bergamaschi, Luca; Putti, Mario International Journal for Numerical Methods in Engineering, Vol. 45, Issue 8 https://doi.org/10.1002/(SICI)1097-0207(19990720)45:8<1025::AID-NME615>3.0.CO;2-G	journal	July 1999
A Comparative Study on Methods for Convergence Acceleration of Iterative Vector Sequences Eyert, V. Journal of Computational Physics, Vol. 124, Issue 2 https://doi.org/10.1006/jcph.1996.0059	journal	March 1996
Orthogonal collocation and alternating-direction procedures for unsaturated flow problems Celia, Michael A.; Ahuja, Lajpat R.; Pinder, George F. Advances in Water Resources, Vol. 10, Issue 4 https://doi.org/10.1016/0309-1708(87)90027-3	journal	December 1987
A 2D high-β Hall MHD implicit nonlinear solver Chacón, L.; Knoll, D. A. Journal of Computational Physics, Vol. 188, Issue 2 https://doi.org/10.1016/S0021-9991(03)00193-1	journal	July 2003
Accurate and economical solution of the pressure-head form of Richards' equation by the method of lines Tocci, Michael D.; Kelley, C. T.; Miller, Cass T. Advances in Water Resources, Vol. 20, Issue 1 https://doi.org/10.1016/S0309-1708(96)00008-5	journal	February 1997
An accelerated Picard method for nonlinear systems related to variably saturated flow Lott, P. A.; Walker, H. F.; Woodward, C. S. Advances in Water Resources, Vol. 38 https://doi.org/10.1016/j.advwatres.2011.12.013	journal	March 2012
Jacobian-free Newton–Krylov methods: a survey of approaches and applications Knoll, D. A.; Keyes, D. E. Journal of Computational Physics, Vol. 193, Issue 2 https://doi.org/10.1016/j.jcp.2003.08.010	journal	January 2004
A monotone finite volume method for advection–diffusion equations on unstructured polygonal meshes Lipnikov, K.; Svyatskiy, D.; Vassilevski, Y. Journal of Computational Physics, Vol. 229, Issue 11 https://doi.org/10.1016/j.jcp.2010.01.035	journal	June 2010
Second-order accurate monotone finite volume scheme for Richards’ equation Misiats, Oleksandr; Lipnikov, Konstantin Journal of Computational Physics, Vol. 239 https://doi.org/10.1016/j.jcp.2012.09.004	journal	April 2013
Nonlinear Krylov acceleration applied to a discrete ordinates formulation of the k-eigenvalue problem Calef, Matthew T.; Fichtl, Erin D.; Warsa, James S. Journal of Computational Physics, Vol. 238 https://doi.org/10.1016/j.jcp.2012.12.024	journal	April 2013
The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient Lipnikov, Konstantin; Manzini, Gianmarco; Moulton, J. David Journal of Computational Physics, Vol. 305 https://doi.org/10.1016/j.jcp.2015.10.031	journal	January 2016
Monotonicity of the cell-centred triangular MPFA method for saturated and unsaturated flow in heterogeneous porous media Younes, Anis; Fahs, Marwan; Belfort, Benjamin Journal of Hydrology, Vol. 504 https://doi.org/10.1016/j.jhydrol.2013.09.041	journal	November 2013
On equivalent hydraulic conductivity for oscillation–free solutions of Richard’s equation Belfort, Benjamin; Younes, Anis; Fahs, Marwan Journal of Hydrology, Vol. 505 https://doi.org/10.1016/j.jhydrol.2013.09.047	journal	November 2013
A Finite-Volume Method with Hexahedral Multiblock Grids for Modeling Flow in Porous Media Lee, S. H.; Jenny, P.; Tchelepi, H. A. Computational Geosciences, Vol. 6, Issue 3/4, p. 353-379 https://doi.org/10.1023/A:1021287013566	journal	January 2002
A general mass-conservative numerical solution for the unsaturated flow equation Celia, Michael A.; Bouloutas, Efthimios T.; Zarba, Rebecca L. Water Resources Research, Vol. 26, Issue 7 https://doi.org/10.1029/WR026i007p01483	journal	July 1990
Capillary Conduction of Liquids Through Porous Mediums Richards, L. A. Physics, Vol. 1, Issue 5 https://doi.org/10.1063/1.1745010	journal	November 1931
A class of methods for solving nonlinear simultaneous equations Broyden, C. G. Mathematics of Computation, Vol. 19, Issue 92 https://doi.org/10.1090/S0025-5718-1965-0198670-6	journal	January 1965
Database-Related Accuracy and Uncertainty of Pedotransfer Functions Schaap, Marcel G.; Leij, Feike J. Soil Science, Vol. 163, Issue 10 https://doi.org/10.1097/00010694-199810000-00001	journal	January 1998
Inexact Newton Methods Dembo, Ron S.; Eisenstat, Stanley C.; Steihaug, Trond SIAM Journal on Numerical Analysis, Vol. 19, Issue 2 https://doi.org/10.1137/0719025	journal	April 1982
Anderson Acceleration for Fixed-Point Iterations Walker, Homer F.; Ni, Peng SIAM Journal on Numerical Analysis, Vol. 49, Issue 4 https://doi.org/10.1137/10078356X	journal	January 2011
Design and Application of a Gradient-Weighted Moving Finite Element Code I: in One Dimension Carlson, Neil N.; Miller, Keith SIAM Journal on Scientific Computing, Vol. 19, Issue 3 https://doi.org/10.1137/S106482759426955X	journal	May 1998
NITSOL: A Newton Iterative Solver for Nonlinear Systems Pernice, Michael; Walker, Homer F. SIAM Journal on Scientific Computing, Vol. 19, Issue 1 https://doi.org/10.1137/S1064827596303843	journal	January 1998
A Unified Approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume Methods Droniou, JÉRÔMe; Eymard, Robert; GallouËT, Thierry Mathematical Models and Methods in Applied Sciences, Vol. 20, Issue 02 https://doi.org/10.1142/S0218202510004222	journal	February 2010
M-Adaptation in the mimetic finite difference method Gyrya, Vitaliy; Lipnikov, Konstantin; Manzini, Gianmarco Mathematical Models and Methods in Applied Sciences, Vol. 24, Issue 08 https://doi.org/10.1142/S0218202514400053	journal	May 2014
Iterative Procedures for Nonlinear Integral Equations Anderson, Donald G. Journal of the ACM, Vol. 12, Issue 4 https://doi.org/10.1145/321296.321305	journal	October 1965
Minimal stencil finite volume scheme with the discrete maximum principle Lipnikov, K.; Svyatskiy, D.; Vassilevski, Yu. Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 27, Issue 4 https://doi.org/10.1515/rnam-2012-0020	journal	January 2012
A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils1 van Genuchten, M. Th. Soil Science Society of America Journal, Vol. 44, Issue 5 https://doi.org/10.2136/sssaj1980.03615995004400050002x	journal	January 1980
Picard-Newton Iterative Method with Time Step Control for Multimaterial Non-Equilibrium Radiation Diffusion Problem Yue, Jingyan; Yuan, Guangwei Communications in Computational Physics, Vol. 10, Issue 4 https://doi.org/10.4208/cicp.310110.161010a	journal	October 2011
Three-phase numerical model for subsurface hydrology in permafrost-affected regions (PFLOTRAN-ICE v1.0) Karra, S.; Painter, S. L.; Lichtner, P. C. The Cryosphere, Vol. 8, Issue 5 https://doi.org/10.5194/tc-8-1935-2014	journal	January 2014
Methods of conjugate gradients for solving linear systems Hestenes, M. R.; Stiefel, E. Journal of Research of the National Bureau of Standards, Vol. 49, Issue 6 https://doi.org/10.6028/jres.049.044	journal	December 1952

Cited By (1)

Numerical Solution of Richards' Equation: A Review of Advances and Challenges Farthing, Matthew W.; Ogden, Fred L. Soil Science Society of America Journal, Vol. 81, Issue 6 https://doi.org/10.2136/sssaj2017.02.0058	journal	October 2017

Similar Records

Application of the Jacobian-Free Newton-Krylov Method to Nonlinear Acceleration of Transport Source Iteration in Slab Geometry

Journal Article · Mon Jan 31 23:00:00 EST 2011 · Nuclear Science and Engineering · OSTI ID:1016204

Parallel-in-Time Solution of Hyperbolic PDE Systems via Characteristic-Variable Block Preconditioning

Journal Article · Wed Oct 08 00:00:00 EDT 2025 · SIAM Journal on Scientific Computing · OSTI ID:3001304

Preconditioning Newton-Krylor Methods for Variably Saturated Flow

Conference · Thu Jan 06 23:00:00 EST 2000 · OSTI ID:791027

Related Subjects

58 GEOSCIENCES
97 MATHEMATICS AND COMPUTING
Mathematics
Planetary Sciences
Richards' equation
variable saturated flow
nonlinear solver
Picard method

New preconditioning strategy for Jacobian-free solvers for variably saturated flows with Richards’ equation

Citation Formats

References (30)

Cited By (1)

Similar Records

Related Subjects