Local radial basis function based gridfree scheme for unsteady incompressible viscous flows
Journal of Computational Physics
Multi-level meshless methods based on direct multi-elliptic interpolation
Journal of Computational and Applied Mathematics
The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems
Journal of Computational Physics
Multicloud: Multigrid convergence with a meshless operator
Journal of Computational Physics
RBF-FD formulas and convergence properties
Journal of Computational Physics
A localized approach for the method of approximate particular solutions
Computers & Mathematics with Applications
Multiscale integrated numerical simulation approach in Štore - steel casthouse
ICOSSE'06 Proceedings of the 5th WSEAS international conference on System science and simulation in engineering
Optimal constant shape parameter for multiquadric based RBF-FD method
Journal of Computational Physics
Stable Computations with Gaussian Radial Basis Functions
SIAM Journal on Scientific Computing
Optimal variable shape parameter for multiquadric based RBF-FD method
Journal of Computational Physics
Journal of Computational Physics
Local method of approximate particular solutions for two-dimensional unsteady Burgers' equations
Computers & Mathematics with Applications
Super linear speedup in a local parallel meshless solution of thermo-fluid problems
Computers and Structures
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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.