Volume 1, Issue 5
Multiquadric Finite Difference (MQ-FD) Method and its Application

Yong Yuan Shan, Chang Shu & Ning Qin

Adv. Appl. Math. Mech., 1 (2009), pp. 615-638.

Published online: 2009-01

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  • Abstract

The conventional finite difference (FD) schemes are based on the low order polynomial approximation in a local region. This paper shows that when the polynomial approximation is replaced by the multiquadric (MQ) function approximation in the same region, a new FD method, which is termed as MQ-FD method in this work, can be developed. The paper gives analytical formulas of the MQ-FD method and carries out a performance study for its derivative approximation and solution of Poisson equation and the incompressible Navier-Stokes equations. In addition, the effect of the shape parameter c in MQ on the formulas of the MQ-FD method is analyzed. Derivative approximation in one-dimensional space and Poisson equation in two-dimensional space are taken as model problems to study the accuracy of the MQ-FD method. Furthermore, a lid-driven flow problem in a square cavity is simulated by the MQ-FD method. The obtained results indicate that this method may solve the engineering problem very accurately with a proper choice of the shape parameter c.

  • Keywords

MQ--FD method shape parameter central FD method

  • AMS Subject Headings

41A10 41A30 65N05

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COPYRIGHT: © Global Science Press

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@Article{AAMM-1-615, author = {Yong Yuan Shan, Chang Shu and Ning Qin}, title = {Multiquadric Finite Difference (MQ-FD) Method and its Application}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2009}, volume = {1}, number = {5}, pages = {615--638}, abstract = {

The conventional finite difference (FD) schemes are based on the low order polynomial approximation in a local region. This paper shows that when the polynomial approximation is replaced by the multiquadric (MQ) function approximation in the same region, a new FD method, which is termed as MQ-FD method in this work, can be developed. The paper gives analytical formulas of the MQ-FD method and carries out a performance study for its derivative approximation and solution of Poisson equation and the incompressible Navier-Stokes equations. In addition, the effect of the shape parameter c in MQ on the formulas of the MQ-FD method is analyzed. Derivative approximation in one-dimensional space and Poisson equation in two-dimensional space are taken as model problems to study the accuracy of the MQ-FD method. Furthermore, a lid-driven flow problem in a square cavity is simulated by the MQ-FD method. The obtained results indicate that this method may solve the engineering problem very accurately with a proper choice of the shape parameter c.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.09-m0942}, url = {http://global-sci.org/intro/article_detail/aamm/8388.html} }
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