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Title: 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:
ORCiD logo [1];  [1]
  1. 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}
}

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Cited by: 6 works
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Works referencing / citing this record:

Review of numerical solution of Richardson–Richards equation for variably saturated flow in soils
journal, June 2019

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