Optimal constant shape parameter for multiquadric based RBF-FD method

Authors:
Victor Bayona;Miguel Moscoso;Manuel Kindelan
Affiliations:
Gregorio Millán Institute, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain;Gregorio Millán Institute, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain;Gregorio Millán Institute, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
Venue:
Journal of Computational Physics
Year:
2011

Citing 15
Cited 4

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Abstract

Radial basis functions (RBFs) have become a popular method for interpolation and solution of partial differential equations (PDEs). Many types of RBFs used in these problems contain a shape parameter, and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter. In this paper, we focus on PDE problems solved with a multiquadric based RBF finite difference (RBF-FD) method. We propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error. The algorithm is based on analytical approximations to the local RBF-FD error derived in [1]. We show through several examples in 1D and 2D, both with structured and unstructured nodes, that very accurate solutions (compared to finite differences) can be achieved using the optimal value of the constant shape parameter.