A new higher-order RBF-FD scheme with optimal variable shape parameter for partial differential equation

Y. L. Ng, Khai Ching Ng, T. W.H. Sheu

Research output: Contribution to journalArticle

Abstract

Radial basis functions (RBFs) with multiquadric (MQ) kernel have been commonly used to solve partial differential equation (PDE). The MQ kernel contains a user-defined shape parameter (ε), and the solution accuracy is strongly dependent on the value of this ε. In this study, the MQ-based RBF finite difference (RBF-FD) method is derived in a polynomial form. The optimal value of ε is computed such that the leading error term of the RBF-FD scheme is eliminated to improve the solution accuracy and to accelerate the rate of convergence. The optimal ε is computed by using finite difference (FD) and combined compact differencing (CCD) schemes. From the analyses, the optimal ε is found to vary throughout the domain. Therefore, by using the localized shape parameter, the computed PDE solution accuracy is higher as compared to the RBF-FD scheme which employs a constant value of ε. In general, the solution obtained by using the ε computed from CCD scheme is more accurate, but at a higher computational cost. Nevertheless, the cost-effectiveness study shows that when the number of iterative prediction of ε is limited to two, the present RBF-FD with ε by CCD scheme is as effective as the one using FD scheme.

Original languageEnglish
Pages (from-to)289-311
Number of pages23
JournalNumerical Heat Transfer, Part B: Fundamentals
Volume75
Issue number5
DOIs
Publication statusPublished - 04 May 2019

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cost effectiveness
Cost-effectiveness
Shape Parameter
Cost effectiveness
Iterative methods
Radial Functions
Finite Difference Scheme
Finite difference method
partial differential equations
Partial differential equations
Basis Functions
Partial differential equation
Higher Order
Iteration
Finite Difference
kernel
Error term
Difference Method
Accelerate
Computational Cost

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modelling and Simulation
  • Condensed Matter Physics
  • Mechanics of Materials
  • Computer Science Applications

Cite this

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abstract = "Radial basis functions (RBFs) with multiquadric (MQ) kernel have been commonly used to solve partial differential equation (PDE). The MQ kernel contains a user-defined shape parameter (ε), and the solution accuracy is strongly dependent on the value of this ε. In this study, the MQ-based RBF finite difference (RBF-FD) method is derived in a polynomial form. The optimal value of ε is computed such that the leading error term of the RBF-FD scheme is eliminated to improve the solution accuracy and to accelerate the rate of convergence. The optimal ε is computed by using finite difference (FD) and combined compact differencing (CCD) schemes. From the analyses, the optimal ε is found to vary throughout the domain. Therefore, by using the localized shape parameter, the computed PDE solution accuracy is higher as compared to the RBF-FD scheme which employs a constant value of ε. In general, the solution obtained by using the ε computed from CCD scheme is more accurate, but at a higher computational cost. Nevertheless, the cost-effectiveness study shows that when the number of iterative prediction of ε is limited to two, the present RBF-FD with ε by CCD scheme is as effective as the one using FD scheme.",
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A new higher-order RBF-FD scheme with optimal variable shape parameter for partial differential equation. / Ng, Y. L.; Ng, Khai Ching; Sheu, T. W.H.

In: Numerical Heat Transfer, Part B: Fundamentals, Vol. 75, No. 5, 04.05.2019, p. 289-311.

Research output: Contribution to journalArticle

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