A Comparative Study of Global and Local Meshless Methods for Diffusion-Reaction Equation

Abstract

This paper focuses on the comparative study of global and local mesh- less methods based on collocation with radial basis functions for solving two di- mensional initial boundary value diffusion-reaction problem with Dirichlet and Neumann boundary conditions. A similar study was performed for the boundary value problem with Laplace equation by Lee, Liu, and Fan (2003). In both global and local methods discussed, the time discretization is performed in explicit and implicit way and the multiquadric radial basis functions (RBFs) are used to inter- polate diffusion-reaction variable and its spatial derivatives. Five and nine nodded sub-domains are used in the local support of the local method. Uniform and non- uniform space discretization is used. Accuracy of global and local approaches is assessed as a function of the time and space discretizations, and value of the shape parameter. One can observe the convergence with denser nodes and with smaller time-steps in both methods. The global method is prone to instability due to ill- conditioning of the collocation matrix with the increase of the number of the nodes in cases N t 3000. On the other hand, the global method is more stable with re- spect to the time-step length. Numerical tests with and without noise are conducted based on the methodology proposed in Younga, Fana, Hua, and Atluri (2009). The results show larger stability of the local versions of the method in comparison with the global ones. The accuracy of the local method is comparable with the accuracy of the global method. The local method is more efficient because we solve only a small system of equations for each node in explicit case and a sparse system of equations in implicit case. Hence the local method represents a preferable choice to its global counter part.