The RBF partition of unity method for solving the Klein-Gordon equation
- PDF / 7,541,238 Bytes
- 13 Pages / 595.276 x 790.866 pts Page_size
- 57 Downloads / 157 Views
ORIGINAL ARTICLE
The RBF partition of unity method for solving the Klein‑Gordon equation Mohammadreza Ahmadi Darani1 Received: 25 February 2020 / Accepted: 4 September 2020 © The Author(s) 2020
Abstract In this paper, a localized radial basis function (RBF) method is applied to obtain a global approximation of the solution of two dimensional Klein-Gordon equation on a given bounded domain. We use the RBF partition of unity (RBF-PU) method which is based on partitioning the original domain to several patches and using the RBF approximation on each local domain. Low computational cost and well conditioned final linear system are the main advantages of this method comparing with the original (global) RBF techniques. Numerical experiments show that the given problem could be solved successfully with a reasonable accuracy. Keywords RBF methods · Partition of unity · Klein-Gordon equation
1 Introduction In recent decades, the radial basis functions (RBF) approximations have become a powerful and considerable tool for developing meshfree methods in solving partial differential equations (PDEs). They possess some useful properties such as easy implementation in higher dimension, flexibility with respect to geometry, reasonable convergence, etc. Although the global RBF-based methods avoid mesh generation and use only scattered points, they lead to a full and ill-conditioned final linear system. The conditioning of the final system grows as both the distance between the points and RBF shape parameter decrease. This growth is of exponential order for infinitely smooth RBFs and of algebraic order for finitely smooth RBFs. To overcome this problem, the use of local RBF methods has been suggested. For example RBFgenerated finite difference (RBF-FD) and RBF partition of unity (RBF-PU) methods have been developed in this direction. Also the use of compactly supported RBFs (such as Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00366-020-01171-z) contains supplementary material, which is available to authorized users. * Mohammadreza Ahmadi Darani [email protected] 1
Department of Applied Mathematics Faculty of Mathematical Sciences, Shahrekord University, P. O. Box 115, Shahrekord, Iran
Wendland’s functions [33, 35]) is recommendable. However, to get the convergence the support size of the RBF should be fixed independent of the disctretization which again leads to ill-conditioned interpolation matrices for large number of nodal points. The use of multiscale methods may resolve this problem at a higher computational cost [14, 28, 36]. Also the domain decomposition method could be helpful in this case but implementing this method with optimal subdomains may be highly problem-dependent [32]. The PU technique was introduced by Shepard [29] for reconstructing a function from its scattered values. This method is based on localizing the approximation by partitioning the original domain to small sub domains. In [23], the partition of unity finite element method was introduc
Data Loading...
