Abstract
In this paper, a novel meshless method for the transient modeling of subsurface flow in unsaturated soils was developed. A linearization process for the nonlinear Richards equation using the Gardner exponential model to analyze the transient flow in the unsaturated zone was adopted. For the transient modeling, we proposed a pioneering work using the collocation Trefftz method and utilized the coordinate system in Minkowski spacetime instead of that in the original Euclidean space. The initial value problem for transient modeling of subsurface flow in unsaturated soils can then be transformed into the inverse boundary value problem. A numerical solution obtained in the spacetime coordinate system was approximated by superpositioning Trefftz basis functions satisfying the governing equation for boundary collocation points on partial problem domain boundary in the spacetime coordinate system. As a result, the transient problems can be solved without using the traditional time-marching scheme. The validity of the proposed method is established for several test problems. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stability for the transient modeling of subsurface flow in unsaturated soils.
Highlights
Increasing interest has been shown in recent years in understanding the behavior of unsaturated soils
We proposed a pioneering work using the collocation Trefftz method (CTM) for transient modeling of subsurface flow in unsaturated soils
The initial value problem for transient modeling of subsurface flow in unsaturated soils can be transformed into the inverse boundary value problem
Summary
Increasing interest has been shown in recent years in understanding the behavior of unsaturated soils. Since the Richards equation is highly nonlinear and cannot directly provide an analytical solution, modeling flow process in unsaturated soils is usually based on the numerical solutions of the Richards equation [9,10,11,12,13,14,15]. Water 2017, 9, 954 geometry are generally intractable analytically [28] For such problems, the use of numerical methods, especially the boundary-type meshless method, to obtain approximate solutions is advantageous [29]. The Trefftz method is probably one of the most popular boundary-type meshless methods for solving boundary value problems where approximate solutions are expressed as a linear combination of functions automatically satisfying governing equations [37,38]. Application examples of the steady-state, the one-dimensional and two-dimensional transient problems of subsurface flow in unsaturated soils were carried out
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