Optimal constant shape parameter for multiquadric based RBF-FD method Articles uri icon

publication date

  • August 2011

start page

  • 7384

end page

  • 7399

issue

  • 19

volume

  • 230

international standard serial number (ISSN)

  • 0021-9991

electronic international standard serial number (EISSN)

  • 1090-2716

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.