The meshless methods for numerical solution of the nonlinear Klein-Gordon equation

In this paper, we develop the numerical solution of nonlinear Klein-Gordon equation (NKGE) using the meshless methods. The finite difference scheme and the radial basis functions (RBFs) collocation methods are used to discretize time derivative and spatial derivatives, respectively. Numerical results are given to confirm the accuracy and efficiency of the presented schemes.Publisher's Versio

Meshless Collocation Methods Applied to Problems with Material Discontinuities

The paper deals with the use of a kind of meshless method to solve the problem with material discontinuity on an interface. Such a problem is described by a differential equation with discontinuous coefficients. To solve the problem, the abovementioned method is associated with the subdomain approach that divides the whole domain into subdomains, in which the problem is continuous. To accurately address the analyzed problem, proper continuity conditions are imposed on the interface. The Gaussian kernel, which belongs to the family of infinitely smooth radial basis functions, is taken into consideration as the basis function for the method. It is known that this type of method can provide very high rate of convergence and high accuracy but it suffers from instability. To avoid the instability, some recent advances in kernel methods, based on Mercer’s theorem, are involved in the present paper. The usefulness of the approaches are shown by benchmark problems described by ordinary as well as partial differential equations