| 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 $c$. |
