Mathematical Problems in Engineering

Volume 2018, Article ID 8298915, 19 pages

https://doi.org/10.1155/2018/8298915

## Hermite Radial Basis Collocation Method for Unsaturated Soil Water Movement Equation

^{1}School of Sciences, Xi’an University of Technology, Xi’an, Shaanxi 710054, China^{2}Institute of Water Resources and Hydro-Electric Engineering, Xi’an University of Technology, Xi’an, Shaanxi 710048, China

Correspondence should be addressed to Lijun Su; moc.361@11nujls

Received 24 November 2017; Accepted 4 March 2018; Published 8 April 2018

Academic Editor: Gisele Mophou

Copyright © 2018 Jiao Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Due to the nonlinear diffusion term, it is hard to use the collocation method to solve the unsaturated soil water movement equation directly. In this paper, a nonmesh Hermite collocation method with radial basis functions was proposed to solve the nonlinear unsaturated soil water movement equation with the Neumann boundary condition. By preprocessing the nonlinear diffusion term and using the Hermite radial basis function to deal with the Neumann boundary, the phenomenon that the collocation method cannot be used directly is avoided. The numerical results of unsaturated soil moisture movement with Neumann boundary conditions on the regular and nonregular regions show that the new method improved the accuracy significantly, which can be used to solve the low precision problem for the traditional collocation method when simulating the Neumann boundary condition problem. Moreover, the effectiveness and reliability of the algorithm are proved by the one-dimensional and two-dimensional engineering problem of soil water infiltration in arid area. It can be applied to engineering problems.

#### 1. Introduction

Unsaturated soil moisture movement [1] referring to the water in the soil is not filled with all the pores of the water movement, and it is an important fluid movement form of porous media. It has important practical significance in extensive engineering sciences such as runoff calculation, groundwater resource evaluation, soil science, hydrology, farmland irrigation and drainage, and soil salinization prevention and management [2]. At present, the soil water movement equation mostly refers to the Richards’ equation, which is established based on the Buckingham-Darcy law and continuous equation. The specific expression with water content as dependent variable is shown as follows:

In recent years, differential method and finite element method have been used to solve this problem; for example, Lei and Yang [3] used the Ritz finite element method to solve the one-dimensional unsaturated soil water flow problem, used finite difference method to establish a numerical model for the one-dimensional flow of unsaturated soil water, and applied it to practical problems [4] successfully; Ma et al. [5] used ADI alternating implicit difference dispersion and Gauss-Seidel iteration to solve the soil water movement equation; Li et al. [6] proposed a radial basis collocation method, combined with the radial basis function and collocation method, for the one-dimensional soil water movement equation; Su et al. [7] solved the two-dimensional unsaturated soil water movement equation by the radial basis collocation with difference method; Hu et al. [8] solved the boundary value problem by weighted radial basis point method. Many scholars [9–20] have solved a series of the numerical solution problems based on the radial basis point method.

Compared with the finite element method, the meshless method [21, 22] has the following main advantages: no grid, preprocessing, and simple adaptive analysis. It is easy to expand the low-dimensional problem to the high-dimensional problem and solve the problem that is difficult to solve by the traditional numerical methods. Therefore, it was a hot research at home and abroad in last twenty years and considered as a very promising numerical analysis method. Radial basis function meshless method [23] has two main forms: collocation method (strong form), Galerkin method (weak form). The radial basis function collocation method (RBCM) is a meshless method for solving the numerical solution of partial differential equations. The shape function formed by the interpolation of the radial basis functions (RBFs) satisfies the Kronecker Delta function property, so that the essential boundary condition can be applied very well [9]. For the high order boundary value problem, the Neumann boundary condition affects the numerical calculation result directly. However, the research results in [10–20] showed that the accuracy obtained by using direct collocation scheme is a bit poor especially on boundary. Therefore, in order to satisfy the Neumann boundary condition accurately, the essential boundary condition contains the description of the field function and the description of the derivative of the field function.

Approximation of RBFs possesses optimal error estimation and convergence. However, the accuracy of the derivative of the interpolating function is usually very poor on the boundary when collocation method is used. In order to enhance the accuracy of the derivative of the approximation functions, Chu et al. [24] used Hermite radial basis point interpolation method for the nonuniform material gradient plane plate vibration; a meshless method was developed based on the collocation method and ridge basis function interpolation by Wang et al. [25]; Xing [9] constructed the governing equations of laminated plates with large deflection bending problem, approximated these field variables with RBF and HRBF, and discreted the control equations with least square collocation method; Wang et al. [26] proposed an improved meshless method which neither needs the computation of integrals nor requires a partition of the region and its boundary, and this method is applied to elliptic equations for examining its appropriateness; Parzlivand and Shahrezaee [27] mentioned a numerical technique which is a combination of collocation method, radial basis functions, the operational matrix of derivative for radial basis functions, and the new computational technique. Then they got the solution of a parabolic partial differential equation with a time-dependent coefficient subject to an extra measurement; Krowiak [28] proposed a Hermite radial basis function differential quadrature method for higher order equations and so on [29–32].

The traditional radial basis point method has a large error at the Neumann boundary condition, so it cannot be used to deal with the Neumann boundary condition well. However, due to the existence of partial guidance in the Hermite radial basis function, it can reduce the error better. In this paper, the Hermite radial basis function combined with the collocation method was used to solve the unsaturated soil water movement equation with Neumann boundary condition. The Hermite radial basis function interpolation not only is satisfied with the Kronecker Delta function, but also can improve the precision of the interpolation function significantly. In addition, due to the nonlinear diffusion term of the second-order partial derivative term in the soil water equation, the chain rule was used to preprocess the nonlinear term of (1) to avoid the phenomenon that the collocation method cannot be used directly. The remainder of this paper is organized as follows: In Section 2, some preliminaries were provided regarding the Hermite radial basis collocation and nonlinear term processing methods. In Section 3, we introduced the Hermite radial basis collocation scheme for the unsaturated soil water movement equation with Neumann boundary condition. In Section 4, the numerical examples were used to examine the appropriateness and efficiency of the method proposed in this paper by comparing it with some traditional methods and applying it to the real background engineering problem. Conclusions are given at the end of the paper.

#### 2. Hermite Radial Basis Collocation Method (HRBCM)

##### 2.1. Radial Basis Function

RBF [33–37] is also called a distance basis function, which is a function of the distance from the node to the node . The RBFs, which have the advantages of having a simple form, being independent of space dimension and isotropy, and being suitable for applying to the numerical approximation. In general, the RBFs can be divided into two categories. One is compacted supported radial basis function, and the other one is the global radial basis function. In contrast to the compacted supported function, the global radial basis function has better accuracy in the interpolation calculation, but the computational complexity is larger. It is a function that depends on the radial coordinate :where is the Euclidean distance from node to node . Typical RBFs include multiquadrics function: , inverse multiquadrics function: , Gaussians function: , TPS (thin-plate splines): ,

where is a constant () and is an integer.

##### 2.2. Hermite Radial Basis Function Approximation

Using the radial basis collocation method (RBCM), the derivative of the approximation function will produce a large error at the boundary of the region. Thus, RBCM cannot be applied to solve the differential equation with Neumann boundary condition, but HRBCM can be used to solve this problem well [22]. Hermite interpolation of the field quantity function in the local support region near the point is approximated by the RBFs and its derivative at the Neumann boundary which can be expressed as [22]wherewhere is the number of nodes in the local support domain and is the number of nodes on the Neumann boundary; let ; represents the total number of nodes. is the interior points, is the boundary point, is the RBFs, is the undetermined coefficient of the corresponding RBFs, and is the number of undetermined coefficients corresponding to the derivatives of the RBFs on the Neumann boundary. and denoted the direction cosine in and directions on the Neumann boundary, respectively.

We can obtain algebraic equations by interpolating at the* k*th point in the local support domain:where , .

We also can be obtain algebraic equation by interpolating at the* m*th point on the Neumann boundaries:

As shown the Figure 1, “0” is a collocation point, “1”, “2,” and “3” are the Neumann boundary points, “4” and “5” are the Dirichlet boundary points, and the “hollow” points in the circle are the other collocation points used to construct the approximating function at the “0” point.