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

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.

41A10, 41A30, 65N05

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@Article{AAMM-1-615, author = {Yong Yuan and Shan and and 20555 and and Yong Yuan Shan and Shu and Chang and and 20556 and and Shu Chang and Ning and Qin and and 20557 and 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} }
TY - JOUR T1 - Multiquadric Finite Difference (MQ-FD) Method and Its Application AU - Shan , Yong Yuan AU - Chang , Shu AU - Qin , Ning JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 615 EP - 638 PY - 2009 DA - 2009/01 SN - 1 DO - http://doi.org/10.4208/aamm.09-m0942 UR - https://global-sci.org/intro/article_detail/aamm/8388.html KW - MQ–FD method, shape parameter, central FD method. AB -

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 $c$.

Yong Yuan Shan, Chang Shu & Ning Qin. (1970). Multiquadric Finite Difference (MQ-FD) Method and Its Application. Advances in Applied Mathematics and Mechanics. 1 (5). 615-638. doi:10.4208/aamm.09-m0942
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