This paper formulates a simple explicit local version of the classical meshless radial basis function collocation (Kansa) method. The formulation copes with the diffusion equation, applicable in the solution of a broad spectrum of scientific and engineering problems. The method is structured on multiquadrics radial basis functions. Instead of global, the collocation is made locally over a set of overlapping domains of influence and the time-stepping is performed in an explicit way. Only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node. The computational effort thus grows roughly linearly with the number of the nodes. The developed approach thus overcomes the principal large-scale problem bottleneck of the original Kansa method. Two test cases are elaborated. The first is the boundary value problem (NAFEMS test) associated with the steady temperature field with simultaneous involvement of the Dirichlet, Neumann and Robin boundary conditions on a rectangle. The second is the initial value problem, associated with the Dirichlet jump problem on a square. The accuracy of the method is assessed in terms of the average and maximum errors with respect to the density of nodes, number of nodes in the domain of influence, multiquadrics free parameter, and timestep length on uniform and nonuniform node arrangements. The developed meshless method outperforms the classical finite difference method in terms of accuracy in all situations except immediately after the Dirichlet jump where the approximation properties appear similar. © 2006 Elsevier Ltd.
Šarler, B., & Vertnik, R. (2006). Meshfree explicit local radial basis function collocation method for diffusion problems. Computers and Mathematics with Applications, 51(8 SPEC. ISS.), 1269–1282. https://doi.org/10.1016/j.camwa.2006.04.013