International Journal of Differential Equations

Volume 2015, Article ID 762034, 8 pages

http://dx.doi.org/10.1155/2015/762034

## Redistribution of Nodes with Two Constraints in Meshless Method of Line to Time-Dependent Partial Differential Equations

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 41335-1914, Rasht 4193822697, Iran

Received 6 July 2015; Accepted 19 October 2015

Academic Editor: Patricia J. Y. Wong

Copyright © 2015 Jafar Biazar and Mohammad Hosami. 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

Meshless method of line is a powerful device to solve time-dependent partial differential equations. In integrating step, choosing a suitable set of points, such as adaptive nodes in spatial domain, can be useful, although in some cases this can cause ill-conditioning. In this paper, to produce smooth adaptive points in each step of the method, two constraints are enforced in Equidistribution algorithm. These constraints lead to two different meshes known as quasi-uniform and locally bounded meshes. These avoid the ill-conditioning in applying radial basis functions. Moreover, to generate more smooth adaptive meshes another modification is investigated, such as using modified arc-length monitor function in Equidistribution algorithm. Influence of them in growing the accuracy is investigated by some numerical examples. The results of consideration of two constraints are compared with each other and also with uniform meshes.

#### 1. Introduction

Meshless methods have gained much attention in the last decade. They are well known for their simplicity and ability in reconstructing multivariate functions from scattered data. Moreover, meshless methods using radial basis functions (RBFs) are powerful methods to solve partial differential equations (PDEs). The initial development in applying RBFs for numerical solution of PDEs is due to the pioneering work of Kansa [1, 2]. He used some collocation nodes to collocate the RBFs. Meshless methods using RBFs have several advantages comparing to finite difference method (FDM) finite element method (FEM) and other mesh based methods [3]. One of these advantages is that they do not require a mesh or element. These methods need only some scattered nodes. It means that the nodes can be chosen, freely. Due to this useful property, an important geometric problem arises: how to choose the nodes to improve the accuracy? This problem causes too many researches about distribution of nodes in meshless methods. Several researchers have considered this problem [4–8]. One of the effective methods to choose an efficient set of central nodes, known as adaptive nodes, is Equidistribution method [9, 10]. In this method, the objective is to find a partition of the interval, such that a given weight function takes a given constant value over each subinterval. These adaptive central nodes can be used in meshless methods, such as meshless method of line (MMOL). The method of line (MOL) is a well-known numerical method to solve PDEs [11, 12]. In the meshless method of line, the RBFs are used to approximate the solution in MOL. This method is very reliable for using adaptive nodes [13, 14]. In each step of this method, some central nodes, in spatial domain, are required. Adaptive central nodes can be a good selection to use in this method. But, due to ill-conditioning of the problem in some cases, when the nodes are near to each other, in many practical cases, it is necessary for the chosen nodes to have certain smoothness properties. This leads to some constraints.

In this study, two reliable constraints are presented, and the influence of them in applying the MMOL with adaptive nodes is investigated. The first constraint, known as quasi-uniform mesh, has been applied in some researches as well [13]. The aim of this study is to show application of another constraint, known as locally bounded mesh, and to discuss impact of it and some modification on meshes on increasing the accuracy. This paper is organized as follows. In Section 2, radial basis functions interpolation is introduced. In Section 3, an Equidistribution algorithm is presented and two constraints are imposed to obtain central nodes. In Section 4, adaptive nodes are applied in MMOL to solve some time-dependent PDEs.

#### 2. Radial Basis Functions to Approximate a Function

Radial basis functions are real valued basis functions which depend on the distance between two points. The commonly used RBFs are multiquadric (MQ), Gaussian, Thin-Plate Spline (TPS), and compactly supported RBFs (CS-RBFs). The MQ radial basis function provides the most accurate approximation, in most applications of RBFs [15]. MQ is defined as , where is called shape parameter. In this paper, MQ is used in numerical examples. To approximate a given scattered data at nodes , RBF interpolation is given by combination of RBFs; that is,where and denote the Euclidean norm. is an RBF and s are the coefficients that will be determined. By collocating the interpolation conditions (, ), the system of equations is obtained as the following matrix form:where and .

The accuracy of the approximate function depends on various factors. Some of the most important ones are as follows: how the RBFs are chosen, nodes distribution, and selecting the shape parameter. Finding the optimal shape parameter is an open problem, although concentrated researches have been made to determine some appropriate shape parameter for a given problem [16, 17]. In this paper, we focus on distributing the nodes to obtain more accurate approximations. In next section, an Equidistribution method is introduced to select adaptive central nodes.

#### 3. An Adaptive Method to Central Nodes

Based on the mesh-free property of RBF meshless methods, one can select a set of nodes freely, such as uniform or random scheme. But in the case that the solution is relatively more oscillatory or even shocks appear, some adaptive schemes can be applied [4–8]. Equidistribution algorithm is a reliable approach to construct adaptive central nodes. Equidistribution is the process of distributing the nodes in an interval such that a determined weight function is equally distributed over the chosen mesh.

##### 3.1. Equidistribution

*Definition 1 (Equidistribution). *Let be a nonnegative piecewise continuous function () and a constant such that is an integer. The meshis called equidistributing (e.d.) on with respect to and , ifThe function is called “monitor,” which depends on the underlying function . To find more details about the monitors refer to [18, 19]. In this paper the arc-length monitor () is applied. To enforce at least a few nodes in the flat part of the interval, a parameter can be inserted in the arc-length monitor, that is, the modified arc-length . In numerical examples the influence of this parameter is illustrated.

For a given monitor function and the constant , the Equidistribution to produce an e.d. mesh in is done in three steps.

*Step 1. *Approximate .

*Step 2. *Determine the smallest integer such that and define .

*Step 3. *Find the mesh by inverse interpolation. The points are given by .

Figure 1 shows the e.d. nodes based on with . It shows that central nodes are concentrated at the region with steepest gradient. The concentration makes the nodes in the region near to each other. With smaller minimum distance between centrals, the MQ shape parameter must be adjusted, so that the condition number of the associated linear system remains reasonable. This adjustment is not always applicable. Thus some constraint on the distribution of central nodes can be imposed. The two most common constraints and an algorithm to construct these constraint adaptive nodes are introduced in the following section.