Space-time meshfree collocation method: Methodology and application to initial-boundary value problems

A novel space-time meshfree collocation method (STMCM) for solving systems of non-linear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh-based methods. The STMCM belongs to the class of truly meshfree methods, i.e. the methods that do not have any underlying mesh, but work on a set of nodes only without any a priori node-to-node connectivity. Instead, the neighbouring information is established on-the-fly. The STMCM is constructed using the Interpolating Moving Least-squares technique, which allows a simplified implementation of boundary conditions due to fulfillment of the Kronecker delta property by the kernel functions, which is not the case for the major part of other meshfree methods. The method is validated by several examples ranging from interpolation problems to the solution of PDEs, whereas the STMCM solutions are compared with either analytical or reference ones. © 2009 John Wiley & Sons, Ltd.