# Construction of Compact Finite Difference Schemes by Classic Differential Quadrature
## 1 Introduction  
Many physical phenomena possess a range of space and time scales, and direct numerical simulations of these processes require all the relevant scales to be properly represented in the numerical model. These requirements have led to the development of the so-called compact finite difference scheme [1]. Usually, the compact schemes have much smaller numerical dispersion and dissipation errors than traditional finite difference schemes of the same order of accuracy on the same mesh. So far, researchers have developed a variety of compact difference schemes [2; 3; 4; 5; 6]. This paper presents a simple way for constructing the compact finite difference schemes by the use of the differential quadrature method (DQM). The DQM was firstly proposed by [PERSON] and [PERSON] in the early 1970s [7; 8]. The basic idea of DQM is to extend analogously the numerical integration rule to the numerical differentiation, where the derivative of a function on a coordinate direction is expressed as the weighted linear combination of function values for all discrete points along this direction. At present, the differential quadrature method has been widely developed. According to the used trial or test function, the differential quadrature method generally includes the classical differential quadrature method, harmonic differential quadrature method, Fourier differential quadrature method, and so on. The classic differential quadrature method mainly uses the general polynomial or Lagrangian interpolation function as a test function, the harmonic differential quadrature method uses the trigonometric function as the trial function, while the Fourier differential quadrature method uses Fourier series expansion as the trial function [9; 10]. Theoretically, the above differential quadrature rules can be used either for spatial discretization or for time discretization. In other words, the differential quadrature method can be used for the solution of both the partial differential equation and the ordinary differential initial problem [11; 12; 13; 14; 15]. In summary, differential quadrature methods have the advantages of high precision, good numerical stability and convenient programming, and have been successfully applied in many fields [16; 17]. Using classic differential quadrature formulae and uniform grids, we systematically constructed a variety of finite difference schemes, and some of these schemes are consistent with the boundary value methods (BVM) proposed by [PERSON] et al. [PERSON] et al. (2008, 2010). Considering that the derived difference schemes enjoy the same stability and accuracy properties with correspondent DQM but have a simpler form of calculation, the derived difference schemes can be regarded as the compact expression of classic DQM. Naturally, these compact difference methods could be used instead of classic differential quadrature methods for engineering applications (K  
2. The Chebyshev grids \[c_{0}=0;\ c_{z}=\frac{1}{2}\bigg{(}1-\cos\bigg{(}\frac{2z-1}{2s-2}\pi\bigg{)} \bigg{)},z\in(1,\ s-1);\ c_{s}=1\] (8)
3. The Chebyshev-Gauss-Lobatto grids \[c_{z}=\frac{1}{2}\Big{(}1-\cos\Big{(}\frac{z}{s}\pi\Big{)}\Big{)},\ z\in(0,\ s)\] (9)
4. The Legendre grids \[c_{0}=0;c_{z},\ z\in(1,\ s)\] are the zero points of \(\frac{d^{s}}{dx^{s}}\big{(}x^{s}(x-1)^{s}\big{)}=0\); and \(c_{s+1}=1\). Take the following ordinary differentia initial value problem as an example \[\left\{ \begin{array}{l}\frac{d}{dx}y\equiv\dot{y}=f(x,y)\\ y(x=0)=y_{0}\end{array} \right.\] (10)  
Supposing \(x_{n}\) and \(x_{n+1}\) represent respectively the starting point and ending point of an arbitrary interval, \(h=x_{n+1}-x_{n}\) is the interval length or the so-called stencil length.