Multiquadric Finite Difference (MQ-FD) Method and Its Application (Advances in Applied Mathematics and Mechanics, 2018)
The conventional finite difference (FD) schemes are based on the low order polynomial approximation in a local region. This paper showsthat when the polynomial approximation is replaced by themultiquadric (MQ) function approximation in the same region, a newFD method, which is termed as MQ-FD method in this work, can bedeveloped. The paper gives analytical formulas of the MQ-FD methodand carries out a performance study for its derivative approximationand solution of Poisson equation and the incompressibleNavier-Stokes equations. In addition, the effect of the shapeparameter $c$ in MQ on the formulas of the MQ-FD method is analyzed.Derivative approximation in one-dimensional space and Poissonequation in two-dimensional space are taken as model problems tostudy the accuracy of the MQ-FD method. Furthermore, a lid-drivenflow problem in a square cavity is simulated by the MQ-FD method.The obtained results indicate that this method may solve theengineering problem very accurately with a proper choice of theshape parameter $c$.