A second-order accurate finite volume scheme with the discrete maximum principle for solving Richards’ equation on unstructured meshes
Abstract
Richards’s equation describes steady-state or transient flow in a variably saturated medium. For a medium having multiple layers of soils that are not aligned with coordinate axes, a mesh fitted to these layers is no longer orthogonal and the classical two-point flux approximation finite volume scheme is no longer accurate. Here, we propose new second-order accurate nonlinear finite volume (NFV) schemes for the head and pressure formulations of Richards’ equation. We prove that the discrete maximum principles hold for both formulations at steady-state which mimics similar properties of the continuum solution. The second-order accuracy is achieved using high-order upwind algorithms for the relative permeability. Numerical simulations of water infiltration into a dry soil show significant advantage of the second-order NFV schemes over the first-order NFV schemes even on coarse meshes. Since explicit calculation of the Jacobian matrix becomes prohibitively expensive for high-order schemes due to build-in reconstruction and slope limiting algorithms, we study numerically the preconditioning strategy introduced recently in Lipnikov et al. (2016) that uses a stable approximation of the continuum Jacobian. Lastly, numerical simulations show that the new preconditioner reduces computational cost up to 2–3 times in comparison with the conventional preconditioners.
- Authors:
-
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Publication Date:
- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Biological and Environmental Research (BER)
- OSTI Identifier:
- 1351231
- Alternate Identifier(s):
- OSTI ID: 1416381
- Report Number(s):
- LA-UR-16-28346
Journal ID: ISSN 0309-1708
- Grant/Contract Number:
- AC52-06NA25396
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Advances in Water Resources
- Additional Journal Information:
- Journal Volume: 104; Journal ID: ISSN 0309-1708
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Earth Sciences; Mathematics; Richards equation, maximum principle, finite volume method
Citation Formats
Svyatsky, Daniil, and Lipnikov, Konstantin. A second-order accurate finite volume scheme with the discrete maximum principle for solving Richards’ equation on unstructured meshes. United States: N. p., 2017.
Web. doi:10.1016/j.advwatres.2017.03.015.
Svyatsky, Daniil, & Lipnikov, Konstantin. A second-order accurate finite volume scheme with the discrete maximum principle for solving Richards’ equation on unstructured meshes. United States. https://doi.org/10.1016/j.advwatres.2017.03.015
Svyatsky, Daniil, and Lipnikov, Konstantin. Sat .
"A second-order accurate finite volume scheme with the discrete maximum principle for solving Richards’ equation on unstructured meshes". United States. https://doi.org/10.1016/j.advwatres.2017.03.015. https://www.osti.gov/servlets/purl/1351231.
@article{osti_1351231,
title = {A second-order accurate finite volume scheme with the discrete maximum principle for solving Richards’ equation on unstructured meshes},
author = {Svyatsky, Daniil and Lipnikov, Konstantin},
abstractNote = {Richards’s equation describes steady-state or transient flow in a variably saturated medium. For a medium having multiple layers of soils that are not aligned with coordinate axes, a mesh fitted to these layers is no longer orthogonal and the classical two-point flux approximation finite volume scheme is no longer accurate. Here, we propose new second-order accurate nonlinear finite volume (NFV) schemes for the head and pressure formulations of Richards’ equation. We prove that the discrete maximum principles hold for both formulations at steady-state which mimics similar properties of the continuum solution. The second-order accuracy is achieved using high-order upwind algorithms for the relative permeability. Numerical simulations of water infiltration into a dry soil show significant advantage of the second-order NFV schemes over the first-order NFV schemes even on coarse meshes. Since explicit calculation of the Jacobian matrix becomes prohibitively expensive for high-order schemes due to build-in reconstruction and slope limiting algorithms, we study numerically the preconditioning strategy introduced recently in Lipnikov et al. (2016) that uses a stable approximation of the continuum Jacobian. Lastly, numerical simulations show that the new preconditioner reduces computational cost up to 2–3 times in comparison with the conventional preconditioners.},
doi = {10.1016/j.advwatres.2017.03.015},
journal = {Advances in Water Resources},
number = ,
volume = 104,
place = {United States},
year = {Sat Mar 18 00:00:00 EDT 2017},
month = {Sat Mar 18 00:00:00 EDT 2017}
}
Web of Science
Works referenced in this record:
A compact multipoint flux approximation method with improved robustness
journal, September 2008
- Aavatsmark, I.; Eigestad, G. T.; Mallison, B. T.
- Numerical Methods for Partial Differential Equations, Vol. 24, Issue 5
A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media
journal, June 2009
- Agelas, Léo; Eymard, Robert; Herbin, Raphaèle
- Comptes Rendus Mathematique, Vol. 347, Issue 11-12
Iterative Procedures for Nonlinear Integral Equations
journal, October 1965
- Anderson, Donald G.
- Journal of the ACM, Vol. 12, Issue 4
On equivalent hydraulic conductivity for oscillation–free solutions of Richard’s equation
journal, November 2013
- Belfort, Benjamin; Younes, Anis; Fahs, Marwan
- Journal of Hydrology, Vol. 505
Mixed finite elements and Newton-type linearizations for the solution of Richards' equation
journal, July 1999
- Bergamaschi, Luca; Putti, Mario
- International Journal for Numerical Methods in Engineering, Vol. 45, Issue 8
A general mass-conservative numerical solution for the unsaturated flow equation
journal, July 1990
- Celia, Michael A.; Bouloutas, Efthimios T.; Zarba, Rebecca L.
- Water Resources Research, Vol. 26, Issue 7
A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes
journal, January 2009
- Danilov, A. A.; Vassilevski, Yu. V.
- Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 24, Issue 3
Second-order accurate finite volume method for well-driven flows
journal, February 2016
- Dotlić, M.; Vidović, D.; Pokorni, B.
- Journal of Computational Physics, Vol. 307
A Unified Approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume Methods
journal, February 2010
- Droniou, JÉRÔMe; Eymard, Robert; GallouËT, Thierry
- Mathematical Models and Methods in Applied Sciences, Vol. 20, Issue 02
Construction and Convergence Study of Schemes Preserving the Elliptic Local Maximum Principle
journal, January 2011
- Droniou, Jérôme; Potier, Christophe Le
- SIAM Journal on Numerical Analysis, Vol. 49, Issue 2
A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids
journal, January 2011
- Friis, Helmer A.; Edwards, Michael G.
- Journal of Computational Physics, Vol. 230, Issue 1
A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils1
journal, January 1980
- van Genuchten, M. Th.
- Soil Science Society of America Journal, Vol. 44, Issue 5
On discrete maximum principles for nonlinear elliptic problems
journal, October 2007
- Karátson, János; Korotov, Sergey; Křížek, Michal
- Mathematics and Computers in Simulation, Vol. 76, Issue 1-3
Three-phase numerical model for subsurface hydrology in permafrost-affected regions (PFLOTRAN-ICE v1.0)
journal, January 2014
- Karra, S.; Painter, S. L.; Lichtner, P. C.
- The Cryosphere, Vol. 8, Issue 5
Convergence of multipoint flux approximations on quadrilateral grids
journal, January 2006
- Klausen, Runhild A.; Winther, Ragnar
- Numerical Methods for Partial Differential Equations, Vol. 22, Issue 6
Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés
journal, December 2005
- Le Potier, Christophe
- Comptes Rendus Mathematique, Vol. 341, Issue 12
Hybrid upwind discretization of nonlinear two-phase flow with gravity
journal, August 2015
- Lee, S. H.; Efendiev, Y.; Tchelepi, H. A.
- Advances in Water Resources, Vol. 82
Mimetic finite difference method
journal, January 2014
- Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
- Journal of Computational Physics, Vol. 257
New preconditioning strategy for Jacobian-free solvers for variably saturated flows with Richards’ equation
journal, August 2016
- Lipnikov, Konstantin; Moulton, David; Svyatskiy, Daniil
- Advances in Water Resources, Vol. 94
Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes
journal, February 2009
- Lipnikov, K.; Svyatskiy, D.; Vassilevski, Y.
- Journal of Computational Physics, Vol. 228, Issue 3
A monotone finite volume method for advection–diffusion equations on unstructured polygonal meshes
journal, June 2010
- Lipnikov, K.; Svyatskiy, D.; Vassilevski, Y.
- Journal of Computational Physics, Vol. 229, Issue 11
Minimal stencil finite volume scheme with the discrete maximum principle
journal, January 2012
- Lipnikov, K.; Svyatskiy, D.; Vassilevski, Yu.
- Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 27, Issue 4
An accelerated Picard method for nonlinear systems related to variably saturated flow
journal, March 2012
- Lott, P. A.; Walker, H. F.; Woodward, C. S.
- Advances in Water Resources, Vol. 38
A Second-Order Maximum Principle Preserving Finite Volume Method for Steady Convection-Diffusion Problems
journal, January 2005
- Bertolazzi, Enrico; Manzini, Gianmarco
- SIAM Journal on Numerical Analysis, Vol. 43, Issue 5
Second-order accurate monotone finite volume scheme for Richards’ equation
journal, April 2013
- Misiats, Oleksandr; Lipnikov, Konstantin
- Journal of Computational Physics, Vol. 239
A monotone nonlinear finite volume method for diffusion equations and multiphase flows
journal, December 2013
- Nikitin, Kirill; Terekhov, Kirill; Vassilevski, Yuri
- Computational Geosciences, Vol. 18, Issue 3-4
A monotone nonlinear finite volume method for advection–diffusion equations on unstructured polyhedral meshes in 3D
journal, January 2010
- Nikitin, K.; Vassilevski, Yu.
- Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 25, Issue 4
Monotonicity of control volume methods
journal, January 2007
- Nordbotten, J. M.; Aavatsmark, I.; Eigestad, G. T.
- Numerische Mathematik, Vol. 106, Issue 2
Integrated surface/subsurface permafrost thermal hydrology: Model formulation and proof-of-concept simulations: INTEGRATED PERMAFROST THERMAL HYDROLOGY
journal, August 2016
- Painter, Scott L.; Coon, Ethan T.; Atchley, Adam L.
- Water Resources Research, Vol. 52, Issue 8
Capillary Conduction of Liquids Through Porous Mediums
journal, November 1931
- Richards, L. A.
- Physics, Vol. 1, Issue 5
Database-Related Accuracy and Uncertainty of Pedotransfer Functions
journal, January 1998
- Schaap, Marcel G.; Leij, Feike J.
- Soil Science, Vol. 163, Issue 10
The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes
journal, April 2011
- Sheng, Zhiqiang; Yuan, Guangwei
- Journal of Computational Physics, Vol. 230, Issue 7
On maximum principles for monotone matrices
journal, June 1986
- Stoyan, Gisbert
- Linear Algebra and its Applications, Vol. 78
Anderson Acceleration for Fixed-Point Iterations
journal, January 2011
- Walker, Homer F.; Ni, Peng
- SIAM Journal on Numerical Analysis, Vol. 49, Issue 4
A Multipoint Flux Mixed Finite Element Method
journal, January 2006
- Wheeler, Mary F.; Yotov, Ivan
- SIAM Journal on Numerical Analysis, Vol. 44, Issue 5
Monotone finite volume schemes for diffusion equations on polygonal meshes
journal, June 2008
- Yuan, Guangwei; Sheng, Zhiqiang
- Journal of Computational Physics, Vol. 227, Issue 12
Works referencing / citing this record:
Review of numerical solution of Richardson–Richards equation for variably saturated flow in soils
journal, June 2019
- Zha, Yuanyuan; Yang, Jinzhong; Zeng, Jicai
- Wiley Interdisciplinary Reviews: Water, Vol. 6, Issue 5