Numerical comparison of least square-based finite-difference (LSFD) and radial basis function-based finite-difference (RBFFD) methods

1. I N T R O D U C T I O N In the past decade, the so-called mesh-free methods have become one of the hot tes t research areas in computa t ional mechanics. The te rm mesh-free or meshless is used to describe the special ways of constructing the approximation or interpolation scheme for the spatial discretization. Tha t is, the function and its derivatives at one central node are approximated entirely from the information of a set of scattered nodes within its local support , and there is no prespecified connectivi ty or relationships among tile nodes. The interest in these schemes is instinctively spurred by the perceived difficulty of generating appropriate meshes in the s tandard schemes for problems characterized by complex geometry or *Author to whom all correspondence should be addressed. 0898-1221/06/$ see front matter @ 2006 Elsevier Ltd. All rights reserved. Typeset by AjMS-~X doi:10.1016/j.camwa.2006.04.015 1298 c. SHU et al. complex physics. Instead of mesh generation, mesh-free methods usually require node generation. From the point of view of computational efforts, node generation is considered as an easier and faster job compared to the former one. After generating a set of nodes within the analysis domain, a local cloud of neighboring points, which is called "local support", is selected for the interior node. Then, a local approximation is constructed based on the local support to approximate the function and/or its derivatives in terms of point values. Finally, a set of algebraic equations can be obtained by substituting the approximation scheme into the governing equations. This is a common procedure to discretize partial differential equations by the mesh-free method. In addition, we can also enjoy the computational ease of adding and deleting nodes from the preexisting of nodes. This property is highly appreciated in the flow problems with large deformation or moving boundaries. In the viewpoint of kernel interpolation/approximation techniques, the mesh-free methods to date can be grouped into two categories. One is based on the least square (LS) technique or its equivalents. This interpolation scheme has been adopted by many popular mesh-free methods [1-8]. The least-squares technique allows an optimized approximation derived from an overdetermined set of equations, and generally the resultant coefficient matrix has good properties such as positive, symmetric, and definite. Thus, by means of using more local supporting-points (more than the unknowns), the problem of singular/ill-conditioned coefficient matrix arising from the polynomial interpolation and local point distribution can be ultimately circumvented. Another is based on the radial basis functions (RBFs) interpolant. Employed as base functions for multivariate data interpolation, RBFs have already shown their capability to construct a scheme with favorable properties such as high efficiency and good quality. Madych et al. [9] have shown that the multiquadrie-RBF interpolation scheme converges faster as the dimension increases, and converges exponentially as the density of the nodes increases. Motivated by the above attractive merits, many researchers cast their sights on the development of RBFs-based methods in the past decades and a iot of literatures are available now in this field [10-18]. It is of great interest to make a comparison between the mesh-free schemes based on these two kernel approximation techniques in terms of accuracy and efficiency. In this study, we choose two mesh-free methods, i.e., least square-based finite difference (LSFD) [8] and radial basis functionbased finite difference (RBFFD) method [18], as two examples to examine their performance. It must be noted that the RBFFD method is called a local multiquadrature-differential quadrature (LMQDQ) method in [18] which directly reflects the numerical techniques implemented in the method. However, in this paper we rename it the RBFFD method due to the fact that this method is essentially equivalent to the finite-difference formulation with radial basis functions as the trial functions. Another reason is that the new name is more simple and consistent. For the LSFD method, Ding et at. [8] theoretically and experimentally showed that the use of the least-square technique does not cause a deterioration of the approximation accuracy. In other words, the order of the approximation accuracy remains the same as that obtained by the multidimensional Taylor series expansion. However, the additional points in the local support (more than the unknowns) do not contribute to the accuracy improvement. On the contrary, they will decrease the accuracy due to the increment of h. For the accuracy of RBFFD method, it is very difficult to conduct theoretical analysis at this moment. This is because RBF approximation is completely different from the polynomial approximation. So, the powerfui tool for accuracy analysis, i.e., Taylor series expansion (it is implicitly based on polynomial approximation), cannot be applied in the accuracy analysis of RBFFD method. Nevertheless, the accuracy of RBFFD method has been studied through numerical tests [19], and the error of the second-order derivative approximation yields e ~ © ( ( h / c ) ~ ) , in which h is the mesh size, c the value of free shape parameter in the certain range, and n a positive constant and determined by the number of supporting points. Despite the previous analysis, a direct comparison of the two methods may be more appreciated. In this paper, we put the two methods under the same computational conditions in terms of Numerical Comparison 1299 governing equation, boundary condition, node distribution, and number of supporting points. Therefore, we can examine their performance on the accuracy and efficiency. The numerical examples selected are the Poisson equation and lid-driven cavity flow. Since the analytical solution of the Poisson equation is given, this example can be used to examine the numerical error arising from the spatial discretization. The case of lid-driven cavity flow is a steady flow problem, and it was used to test the performance of two methods in the numerical simulation of complex phenomena. 2. B R I E F D E S C R I P T I O N O F T W O M E S H F R E E M E T H O D S In this section, a brief introduction of the two mesh-free methods is provided. For the details of two methods, one can refer to [8] and [18]. 2.1. Mesh-Free Least Square -Based Fin i te -Di f ference M e t h o d The LSFD method is based on the use of a weighted least square approximation procedure together with a Taylor series expansion of the unknown function. In theory, the multidimensional finite-difference method derived from multidimensional Taylor series expansion can be seen as a natural mesh-fi'ee method since the construction of Taylor series expansion approximation does not require the help of mesh or the connection between the local supporting points. For a smooth function f, suppose that the two-dimensional Taylor series is expanded around one reference node 0 and truncated to the third-order derivative terms. Then, we can obtain a system of equations with nine derivatives as unknowns. It can be written in the matrix form by