Optimal variable shape parameter for multiquadric based RBF-FD method

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...

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Bibliographic Details
Published in:Journal of computational physics 2012-03, Vol.231 (6), p.2466-2481
Main Authors: Bayona, Victor, Moscoso, Miguel, Kindelan, Manuel
Format: Article
Language:English
Subjects:
Accuracy
Approximation
Computational techniques
Dependent variables
Deterioration
Exact sciences and technology
Finite difference method
Mathematical analysis
Mathematical methods in physics
Mathematical models
Mesh-free
Multiquadric
Optimization
Physics
Radial basis functions
Shape parameter
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Summary: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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2011.11.036