### 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 language | English |
---|---|

Pages (from-to) | 289-311 |

Number of pages | 23 |

Journal | Numerical Heat Transfer, Part B: Fundamentals |

Volume | 75 |

Issue number | 5 |

DOIs | |

Publication status | Published - 04 May 2019 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Numerical Heat Transfer, Part B: Fundamentals*,

*75*(5), 289-311. https://doi.org/10.1080/10407790.2019.1627811

}

*Numerical Heat Transfer, Part B: Fundamentals*, vol. 75, no. 5, pp. 289-311. https://doi.org/10.1080/10407790.2019.1627811

**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.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A new higher-order RBF-FD scheme with optimal variable shape parameter for partial differential equation

AU - Ng, Y. L.

AU - Ng, Khai Ching

AU - Sheu, T. W.H.

PY - 2019/5/4

Y1 - 2019/5/4

N2 - 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.

AB - 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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UR - http://www.scopus.com/inward/citedby.url?scp=85067436115&partnerID=8YFLogxK

U2 - 10.1080/10407790.2019.1627811

DO - 10.1080/10407790.2019.1627811

M3 - Article

VL - 75

SP - 289

EP - 311

JO - Numerical Heat Transfer, Part B: Fundamentals

JF - Numerical Heat Transfer, Part B: Fundamentals

SN - 1040-7790

IS - 5

ER -