Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Radial basis functions method for numerical solution of the modified equal width equation

Abstract: The numerical solution of the modified equal width equation is investigated by using meshless method based on collocation with the well-known radial basis functions. Single solitary wave motion, two solitary waves interaction and three solitary waves interaction are studied. Results of the meshless methods with different radial basis functions are presented.

The Midpoint Upwind Finite Difference Scheme for Time-Dependent Singularly Perturbed Convection-Diffusion Equations on Non-Uniform Mesh

Abstract: A numerical approach is proposed to examine the singularly perturbed time dependent convection-diffusion equation in one space dimension on a rectangular domain. The solution of considered problem exhibits a boundary layer on the right side of the domain. We semidiscretize the continuous problem by means of backward Euler finite difference method in the temporal direction. The semi-discretization process yields a set of ordinary differential equations at each time level. A resulting set of ordinary differential equations are discretized by using midpoint upwind finite difference scheme on a non-uniform mesh of Shishkin type. The resulting finite difference method is shown to be almost of second order accurate in the coarse mesh and almost of first order accurate in fine mesh in the spatial direction. First order accuracy is achieved in the temporal direction. An extensive amount of analysis has been carried out in order to obtain uniform convergence of the method. Finally, we have found that the method is...

Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media

Abstract: In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa’s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa’s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.