The Five-Point Difference Method Based on the K-Modes Cluster for Two-Dimensional Poisson Equations

Scientific computation plays an important role in natural sciences, engineering and technology. It has become an indispensable tool in many areas. The Poisson equation is an essential partial differential equation in fluid dynamics and electromagnetic. Due to its analytical solution of a Poisson equation is difficult to obtain, an alternative method is to find its numerical solution. A classical method to get the numerical solution of the Poisson equation is the standard five-point or nine-point difference scheme, which is simple to use and secondorder accurate [1-3]. In order to enhance the accuracy of the difference scheme, it is often required to have finer mesh split or to use more complex difference scheme, which will result in much more computing time and larger amount of data storage. Moreover, as the scale of data increases, the dimension of the coefficient matrix of the linear equations will increase and the numerical solutions will become less stable. To acquire good experimental data usually need a compromise between the calculation efficiency and data storage capacity. In the numerical solution of the two-dimensional Poisson equation, the function value often fluctuates considerably. In areas with a higher absolute value of the derivative (if it exists), the mesh accuracy needs to be higher. In doing so, we need to apply to the whole domain, leading to the data storage and computation time doubled. In fact, in some parts of the domain, even if the division accuracy increases, the accuracy of the numerical solution may stay the same. Therefore, it is of importance to simplify the calculation and to reduce data storage while ensuring sufficient accuracy of numerical solutions.


Introduction
Scientific computation plays an important role in natural sciences, engineering and technology. It has become an indispensable tool in many areas. The Poisson equation is an essential partial differential equation in fluid dynamics and electromagnetic. Due to its analytical solution of a Poisson equation is difficult to obtain, an alternative method is to find its numerical solution. A classical method to get the numerical solution of the Poisson equation is the standard five-point or nine-point difference scheme, which is simple to use and secondorder accurate [1][2][3]. In order to enhance the accuracy of the difference scheme, it is often required to have finer mesh split or to use more complex difference scheme, which will result in much more computing time and larger amount of data storage. Moreover, as the scale of data increases, the dimension of the coefficient matrix of the linear equations will increase and the numerical solutions will become less stable. To acquire good experimental data usually need a compromise between the calculation efficiency and data storage capacity. In the numerical solution of the two-dimensional Poisson equation, the function value often fluctuates considerably. In areas with a higher absolute value of the derivative (if it exists), the mesh accuracy needs to be higher. In doing so, we need to apply to the whole domain, leading to the data storage and computation time doubled. In fact, in some parts of the domain, even if the division accuracy increases, the accuracy of the numerical solution may stay the same. Therefore, it is of importance to simplify the calculation and to reduce data storage while ensuring sufficient accuracy of numerical solutions.
The proposed new k-modes method is an extension of the k-means algorithm [4] for processing large data in data mining. In 2003, Hu and Chen proposed an effective method based on mesh grid for highdimensional data and large data sets [5]. In 2009, an algorithm based on mesh grid and module indicator optimization hierarchical clustering was also proposed [6]. Recent research on the difference scheme for differential equations has mainly applied new results of other research areas to the classical numerical algorithms. For instance, the doublegrid [7] finite difference method for nonlinear elliptic and parabolic equations was given in 2004. Another method of second order elliptic equation with variable coefficients [8] for solving initial boundary problems was developed in 2010. They provided a difference scheme and proved its existence, uniqueness, convergence and stability using energy method. Recent research on the numerical solution of differential equations is mainly focused on applying data processing algorithms to the problem-solving process, such as the Radial Basis Function (RBF) neural network for solving differential equations [9,10]. Another approach is to construct an effective difference scheme for new forms of differential equations obtained from the engineering field or theoretical deduction. For example, in computational fluid dynamics (CFD) field, building an effective calculation method to capture shock is a key part of the study.
To our knowledge, there are not research publications to use k-modes cluster to construct five-point difference scheme for solving a two-dimensional Poisson equation. In this paper, based on the development of new algorithms in data mining, we applied k-modes cluster to the standard five-point difference scheme and proposed a new method for solving two-dimensional Poisson equation where we constructed the five-point difference scheme based on k-modes cluster algorithm. The proposed new method not only ensures the stabilization of solutions of the fully discrete two-dimensional (2-D) Poisson system, but also reduces the computational load and avoids time-consuming calculations, while guaranteeing the sufficient accuracy of its numerical solution. This new method also enhances calculation efficiency.
In the next section, the k-modes cluster algorithm is given and the new five-point difference scheme based on k-modes clustering for 2-dimentional Poisson equations are developed. Given below, some numerical examples are presented to illustrate the errors between the solutions of the proposed and the standard five-point difference scheme, and below section provides conclusions and future research topics.

K-modes method and the new five-point difference scheme based on k-modes cluster for two-dimensional Poisson equations
We consider the Dirichlet boundary problem of two-dimensional Poisson equation x y , are the given functions, ( ) u x, y is the unknown function.

K-modes algorithm:
The k-means clustering algorithm is widely used in data mining. Compared with k-means algorithm, the k-modes algorithm can handle classified attribute-type data where the k-means algorithm cannot [11,12].
The k-modes algorithm uses modes instead of the means in clustering. It is similar to k-means, which renews alternately the cluster center and divides the matrix, till the value of the cost function remains unchanged. The k-modes algorithm takes the dissimilarity measure ( ) ( ) there are s character properties, ( ) of the distance in the k-means algorithm.
In the k-modes algorithm, the smaller the dissimilarity measure, the smaller the distance. The dissimilarity measure between a sample and a cluster center is the number of their different properties. The sample belongs to the cluster center that has the minimum dissimilarity measure.
Use the following function is minimized as the clustering criterion function: can be obtained. Then, begin to cluster with two forms (see Step d) and Step e)). g) Linking all classes after the data processing in accordance with the original order to form a new subdivision. In the new mesh, using a five-point difference scheme, apply different discretization methods at inner points and junction points of the mesh in order to get the new discrete equation and solve it again.

Numerical Experiments
In this section, numerical experiments are conducted in the twodimensional Poisson equations with Dirichlet boundary value problem, giving The exact solution of problem (3) is ( , ) sin( ) π = x u x y e y . In the following, we apply the five-point difference scheme based on k-modes cluster (see above) to solve problem (3). .

In the interval [ ] [ ]
The mesh directly leads to the five-point difference results, namely the linear equations: The value of the nodes are substituted into (4), obtaining Solve the linear equations (5) by Gauss-Seidel iteration.
The results obtained in Step above, such as the exact solutions of the nodes, the difference function values and the absolute values of the error, are shown in Table 1 (the total number of internal nodes are 464).
Cluster based on the coarse grid function b) The class C 2 * includes the remaining regions in the original given area after removing the regions of the class C 1 *.
In order to obtain a more exact solution, we divide the mesh in a finer way. The class of clustering will be handled as following: a) For the regions corresponding to the class C 1 and C 1 *, maintain the original division fineness unchanged. b) For the regions corresponding to the class C 2 and C 2 *, maintain the original division fineness unchanged on the abscissa axis, while reducing the division step size to a half on the vertical axis. It means that there are 814 internal nodes in the grid function cluster and 856 internal nodes in the error function cluster in total.
After the difference subdivision, we still use the five-point difference scheme for internal nodes. At the junction points of two types, , 1 . The equations obtained above are solved by Gauss-Seidel iterative method.
As a result, comparing the numerical solutions of two-dimensional Poisson problems that are calculated by the five-point difference schemes based on grid functions k-modes clustering and error functions k-modes clustering process, we can give the error curves graph of the numerical solution with five-point difference processing based on grid function u u x y k-modes clustering, which is shown in Figure   1 (The unit of error shaft: ×10 -1 ).
The error curves graph of the numerical solution with difference processing based on error function clustering is shown in Figure 2 (The unit of error shaft: ×10 -1 ).

Conclusions
From Figure 1 and Figure 2, we notice that the error between the numerical solutions calculated by the grid function