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.

- Publication Date:

- Report Number(s):
- LA-UR-16-28346

Journal ID: ISSN 0309-1708

- Grant/Contract Number:
- AC52-06NA25396

- Type:
- Accepted Manuscript

- Journal Name:
- Advances in Water Resources

- Additional Journal Information:
- Journal Volume: 104; Journal ID: ISSN 0309-1708

- Publisher:
- Elsevier

- 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)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Earth Sciences; Mathematics; Richards equation, maximum principle, finite volume method

- OSTI Identifier:
- 1351231

- Alternate Identifier(s):
- OSTI ID: 1416381

```
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.,
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. doi:10.1016/j.advwatres.2017.03.015.

```
Svyatsky, Daniil, and Lipnikov, Konstantin. 2017.
"A second-order accurate finite volume scheme with the discrete maximum principle for solving Richards’ equation on unstructured meshes". United States.
doi: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 = {2017},

month = {3}

}