Radial basis function interpolation: numerical and analytical developments
Radial basis function interpolation: numerical and analytical developments
Scattered node compact finite difference-type formulas generated from radial basis functions
Journal of Computational Physics
Local radial basis function based gridfree scheme for unsteady incompressible viscous flows
Journal of Computational Physics
The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems
Journal of Computational Physics
Meshfree explicit local radial basis function collocation method for diffusion problems
Computers & Mathematics with Applications
RBF-FD formulas and convergence properties
Journal of Computational Physics
Adaptive meshless centres and RBF stencils for Poisson equation
Journal of Computational Physics
Stabilization of RBF-generated finite difference methods for convective PDEs
Journal of Computational Physics
Optimal constant shape parameter for multiquadric based RBF-FD method
Journal of Computational Physics
Journal of Computational Physics
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In this follow up paper to our previous study in Bayona et al. (2011) [2], we present a new technique to compute the solution of PDEs with the multiquadric based RBF finite difference method (RBF-FD) using an optimal node dependent variable value of the shape parameter. This optimal value is chosen so that, to leading order, the local approximation error of the RBF-FD formulas is zero. In our previous paper (Bayona et al., 2011) [2] we considered the case of an optimal (constant) value of the shape parameter for all the nodes. Our new results show that, if one allows the shape parameter to be different at each grid point of the domain, one may obtain very significant accuracy improvements with a simple and inexpensive numerical technique. We analyze the same examples studied in Bayona et al. (2011) [2], both with structured and unstructured grids, and compare our new results with those obtained previously. We also find that, if there are a significant number of nodes for which no optimal value of the shape parameter exists, then the improvement in accuracy deteriorates significantly. In those cases, we use generalized multiquadrics as RBFs and choose the exponent of the multiquadric at each node to assure the existence of an optimal variable shape parameter.