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Title: A second-order accurate finite volume scheme with the discrete maximum principle for solving Richards’ equation on unstructured meshes

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.
ORCiD logo [1] ;  [1]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Report Number(s):
Journal ID: ISSN 0309-1708
Grant/Contract Number:
Accepted Manuscript
Journal Name:
Advances in Water Resources
Additional Journal Information:
Journal Volume: 104; Journal ID: ISSN 0309-1708
Research Org:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Biological and Environmental Research (BER) (SC-23)
Country of Publication:
United States
97 MATHEMATICS AND COMPUTING; Earth Sciences; Mathematics; Richards equation, maximum principle, finite volume method
OSTI Identifier:
Alternate Identifier(s):
OSTI ID: 1416381