An Eulerian‐Lagrangian approach with an adaptively corrected method of characteristics to simulate variably saturated water flow

Abstract

A relatively simple method of characteristics is developed to simulate one‐dimensional variably saturated water flow. The method uses the Eulerian‐Lagrangian approach to separate the governing flow equation into “convection” and “diffusion” parts, which are solved with the method of characteristics and the conventional finite element method, respectively. The method of characteristics combines a single‐step reverse particle tracking technique with a correction strategy to ensure accurate mass balances. The correction process is implemented by weighing the calculated convective contribution to the pressure head at each node with the pressure head values of two upstream nodes, using an adaptive weighing factor λ. The value of λ is automatically adjusted by considering the global mass balance at each time step. Numerical experiments for ponded infiltration are presented to illustrate the scheme's performance for situations involving highly nonlinear soil hydraulic properties and extremely dry initial conditions. Results indicate that the proposed method is mass‐conservative, virtually oscillation‐free, and computationally quite efficient. The method is especially effective for simulating highly nonlinear flow scenarios for which traditional finite difference and finite element numerical methods often fail to converge.

A finite-element algorithm for modeling variably saturated flows

“…(11) by the gauss numerical integral method use the conductivity values of gauss points directly. However, values of hydraulic conductivity were evaluated at the half-time level other than at the nth time level by some researchers (Huang et al, 1994), namely, some kind of average of hydraulic conductivity may be considered over a time-step.…”

“…As pointed out by Milly (1985), a mass-conservative solution may also be obtained with the head based formulation if the capacity term is evaluated in an appropriate manner (Huang et al, 1994). A modified head based formulation (Wu and Gao, 1999) is an alternative mass conservative scheme approximating Eq.…”

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

“…The use of the FDM and FEM is reported in the literature most frequently, because they offer computational advantages for solving parabolic-elliptic equations, such as the Richards equation. However, the FDM and FEM often suffer, to some degree, from mass balance errors as well as numerical oscillations, and additional numerical problems may appear when the gravitational term becomes important [19]. This case is frequently encountered in highly permeable soils, where the advective term of Equation (9) becomes important.…”

Sharp 𝐿¹ a posteriori error analysis for nonlinear convection-diffusion problems

“…The method of characteristic originally proposed in [16,31] is widely used to solve convectiondiffusion problems in the finite element community (cf., e.g., [22,1,21,8,23]). Given (X(t))),X(t n ) = x.…”

“…Problems of the type (1.1) model a wide variety of physical phenomena including porous media flow, flow of glaciers, and sedimentation processes [24]. Our motivation comes from the simulation of flow transport through unsaturated porous media which is governed by the so-called Richards equation [2,22], where x = (x 1 , . .…”