# An Improved Differential Quadrature Time Element Method
## 1 Introduction  
The Differential Quadrature Method (DQM) [1] is an effective numerical technique to be used as a global finite difference scheme or generalized collocation method. The study of DQM has been mainly focused on how to select grid points, how to impose initial or boundary conditions, how to determine weighting coefficients, and convergence of solutions. The relevant and representative publications are reviewed below in turn since DQM is applied to the discretization of spatial and temporal domains in the present work. The extremum points of Legendre polynomial in the domain \([-1,\,1]\)[2] and the so-called Gauss-Lobatto-Chebyshev points [3] were taken as unequally aligned node points, and the latter has been widely accepted by far. The efficient ways to deal with boundary conditions are formulating weighting matrix [4, 5] except \(\delta\) method [3, 6], and modifying the weighting coefficient points [5, 7], adding boundary degrees of freedom [8], employing Differential Quadrature Element Method (DQEM) [9], etc. The initial conditions can be applied via some representative ways, including expressing initial conditions as Differential Quadrature (DQ) analog equations at sampling points [2, 10, 11], modifying DQ rule [10, 12], modifying trial functions [13, 14], and modifying weighting coefficient matrix [15, 16]. Notably, the advantage of modified DQ rule method is the accuracy of state variables at the end of time interval, and its accuracy can be improved up to \(2n\) order by using only \(n+1\) sampling points with unconditional stability and non-dissipation. However, the power of the displacement polynomials is the same with that of the velocity, which violates the differential relation between displacement and velocity [10, 17]. [PERSON] and [PERSON] changed coefficients matrix and generalized load vector to simplify the imposition of initial conditions [18]. As for the weighting coefficients, [PERSON] and [PERSON] obtained the first and second order explicit weighting coefficients by differentiating Lagrange polynomials [19; 20]. [PERSON] and [PERSON] derived high-order weighting coefficients from recursion formulae [21]. In addition, [PERSON] and [PERSON] pointed out that DQM convergence speed is equal to that of Fourier series solution while faster than that resulting from Ritz method; moreover, DQM is much simpler than Fourier series, which is easily implemented by coding and applicable to static and dynamic structural analysis [22]. In the past three decades, DQM has developed to an accurate and efficient numerical method for plenty of applications. In solid mechanics, DQM is usually used for plate or shell [23; 24] structure analysis; many examples refer to mechanical behavior of a plate or panel [25; 26], vibration of composite laminated plate [27; 28] considering lower-order shear deformation, nonlinear bending problems of an orthotropic plate [29], nonlinear vibrations of a plate and nonlinear dynamics of viscoelastic beams [27; 30], etc. In fluid mechanics, typical applications include incompressible viscous fluid flow analysis together with hydrodynamics and heat transfer [3], circulation around a circular cylinder at Reynold's number problems for incompressible viscous fluid flow [31; 3], boundary layer equations solutions [32; 33] and coupled heat transfer problems [2; 34; 35]. In 2009, [PERSON] and [PERSON] [36] proposed a Differential Quadrature Finite Element Method (DQFEM) which was motivated by the complexity of imposing boundary conditions in DQM of strong form and the unsymmetrical element matrices in DQEM. In their later study, the explicit forms were presented for the C\({}^{0}\) and C\({}^{1}\) one-dimensional to three-dimensional structural element matrices and load vectors [37]. The innovative idea of discretizing time coordinates through DQ rule were initially given by [PERSON] et al. [1; 38] and then extended to spatial domain.