Second-order accurate finite volume schemes 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. We propose new secondorder 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 [21] that uses a stable approximation of the continuum Jacobian. Numerical simulations show that the new preconditioner reduces computational cost up to 2-3 times in comparison with the conventional preconditioners.

Review of numerical solution of Richardson–Richards equation for variably saturated flow in soils

“…Many approaches are available to discretize the spatial derivatives in RRE (Bergamaschi & Putti, ; Farthing, Kees, & Miller, ), such as finite element method (Simunek et al, ), finite difference method (Dogan & Motz, ), finite volume method (Caviedes‐Voullième, Garcı´a‐Navarro, & Murillo, ; Svyatskiy & Lipnikov, ), finite analytical method (Zhang et al, ), etc. Under atmospheric boundary conditions (i.e., combinations of sinusoidal flux), the magnitudes of the hydraulic gradient, as well as vertical flux variations, generally dampen from the soil surface to deep soils (Bakker & Nieber, ; Dickinson et al, ).…”

Numerical Solution of the Two-Dimensional Richards Equation Using Alternate Splitting Methods for Dimensional Decomposition

“…Numerical tests were performed for a grid with a constant spatial length ∆x = ∆y = 0.025 m. The analytical solution was compared with the results obtained using dimensional splitting algorithms in the form of the Godunov method (G), Equations (11)- (12); the Strang method (S), Equations (14)- (16); and the modified Strang method (mS), Equations (22)- (26). The calculations were carried out for two variants, depending on the sequence in which the 1D equations are solved at each time level.…”

“…Hydraulic parameters of the soil and numerical parameters used in Test 2. For each soil, the computations were performed using the Godunov method (G), Equations (11)- (12); the Strang method (S), Equations (14)- (16); and the alternate modified method (mod), Equations (29)- (35). As in the previous test, the simulations were carried out for two variants of a sequence of splitting (ZX or XZ) in which 1D equations are solved.…”

“…For solving the 2D and 3D Richards equations, a variety of numerical algorithms based on the finite difference (FD) [3][4][5], finite element (FE) [6][7][8], and finite volume (FV) methods [9][10][11][12] have been proposed. Among the aforementioned methods applied for solving multi-dimensional flow problems, it is necessary also to distinguish coupled algorithms, such as the control-volume FD scheme [13], the mixed FE scheme [14][15][16], and the approach based on semi-discretization, in the form of the method of lines (MoL) and transversal method of lines (TMoL) [17].…”

Modelling Unsaturated Flow in Porous Media Using an Improved Picard Iteration Scheme

“…Dolejší et al (2019) proposed an adaptive higher-order space-time discontinuous-Galerkin (hp-STDG) method to obtain highly efficient and accurate solutions to the RE. Svyatskiy and Lipnikov (2017) proposed a second-order-accurate finite volume scheme that solves the RE using high-order upwind algorithms for the relative permeability. Additionally, other advanced numerical methods for solving variable saturated flow problems exhibit high levels of accuracy, computational efficiency, or ease of implementation under certain conditions.…”