Monotonicity Considerations for Saturated--Unsaturated Subsurface Flow

Abstract

It is demonstrated that monotonicity is a sufficient condition to ensure that no new nonphysical local maxima and minima can be produced in the discrete nonlinear unsaturated flow equation. Monotonicity conditions are derived for various types of weighting for the mobility term. The basic discretization is of finite element type, but the results can be extended to finite volume discretizations. Central weightings are only conditionally monotone, while upstream weightings are unconditionally monotone. Sample computations are given for highly heterogeneous problems.

Modelling the Hydraulic Behaviour of Growing Media with the Explicit Finite Volume Solution

“…The upwind discretization is rarely used by hydrogeologists in practical applications, but Forsyth and Kropinsky [22] and Furhmann and Langmach [23] showed that the upwind technique helps to avoid numerical oscillations in the solution. In order to avoid numerical problems, the discretization scheme must be monotonic, and monotonicity can be assured only if the scheme is first-order accurate [24].…”

Locally conservative, stabilized finite element methods for variably saturated flow

“…Along with the similar "chord-slope" technique from [21], this can be considered the standard approach for treating accumulation terms in Richards' equation. Later, [10] demonstrated that mass-lumping combined with appropriate "upwinded" relative permeability evaluation can lead to monotone approximations independent of mesh resolution.…”

“…The fourth and fifth examples are based on the second test problem from [10]. VGM p-s-k relations are again used with four separate media types configured in a simple block pattern.…”

“…Classical FEMs do not perform well for Richards' equation. Most notably, lumping and upwinding schemes are required to prevent significant non-physical oscillations [6,10] and straightforward evaluation of Darcy's law leads to an approximate groundwater velocity field that is not locally conservative over mesh elements. By this we mean that normal fluxes are discontinuous across element boundaries, and the velocity field's divergence over an element does not balance the discrete mass accumulation [11].…”

“…Mass lumping and upwinding are commonly introduced to smear steep transition zones, while postprocessing schemes or mixed FEM formulations have been considered to obtain locally conservative velocity fields [6,10,[12][13][14]. An exhaustive review of available discretization techniques is beyond the scope of this work.…”