# An Improved Differential Quadrature Time Element Method
## 3 Accuracy and Stability  
By virtue of modal decomposition, the linear undamped multi-DOFs system can be equivalently transformed to a set of single-DOF systems; therefore, in the analysis of accuracy and stability of a new method, single-DOF system is used generally. Consider the following single-DOF system  
\[\ddot{x}+(\omega h)^{2}x=0 \tag{33}\]  
where \(\omega\) is the angular frequency, \(h\) is time step size, here an over dot represents the differential with respect to the non-dimensional time \(\tau\ =\ t/h\), \(t\) is the real time. In this case, displacement vector \(\mathbf{x}\) is  
\[\mathbf{x}^{\mathrm{T}}=\left[ \begin{array}{cccccc}x_{1}&x_{2}&x_{3}&\cdots&x_{n }\end{array} \right] \tag{34}\]  
where \(x_{i}\) (\(i=1,\,2,\,\ldots,\,n\)) denote the displacements at time sampling points, and  
\[\mathbf{G}=\mathbf{B}+(wh)^{2}\ \mathbf{E},\mathbf{Z}=\mathbf{x},\mathbf{R}=\mathbf{0} \tag{35}\]  
According to the method of imposing initial conditions in Reference [18], the relationship between the \(n\)-th sampling point and the initial displacement and velocity is  
\[\left[ \begin{array}{c}x_{n}\\ \dot{x}_{n}\end{array} \right]=J\left[ \begin{array}{c}x_{0}\\ \dot{x}_{0}\end{array} \right] \tag{36}\]  
and the elements of the Jacobi matrix \(\mathbf{J}\) are  
\[\left\{ \begin{array}{l}J_{11}=S_{n1}\\ J_{12}=S_{nn}\\ J_{21}=A_{n1}S_{11}+A_{n1}S_{21}+\cdots+A_{nm}S_{n1}\\ J_{22}=A_{n1}S_{1n}+A_{n1}S_{2n}+\cdots+A_{nm}S_{nn}\end{array} \right. \tag{37}\]  
where \(S=G^{-1}\), then stable interval can be obtained by using criterion \(\rho(\mathbf{J})\leq 1\), \(\rho(\mathbf{J})\) is the spectral radius, which is shown in Figure 1, and the first stable intervals are listed in Table 2. It should be stressed that, for different sampling point \(n\), \(\rho(\mathbf{J})>1\) or \(\rho(\mathbf{J})=1\). \begin{table}
\begin{tabular}{c c c c} \hline
**Method** & **Equation** & **Time Domain** & **Spatial Domain** \\ \hline DQTFEM [40] & Integral form or weak form & \begin{tabular}{c} DQ time finite element \\ with non-uniform nodes \\ \(\dot{x}_{i}=A_{i}\mathbf{x}_{i}\) \\ \end{tabular} & \begin{tabular}{c} Conventional finite element \\ with uniform nodes \\ \end{tabular} \\ \hline DQTEM0 [18] & Differential form or strong form & \begin{tabular}{c} DQ time element with \\ non-uniform nodes \\ \(\dot{x}_{i}=A_{i}\mathbf{x}_{i},\ddot{x}_{i}=A_{i}^{2}\mathbf{x}_{i}\) \\ \end{tabular} & \begin{tabular}{c} Conventional finite element \\ with uniform nodes \\ \end{tabular} \\ \hline Present & Differential form or strong form & \
\begin{tabular}{c} DQ time element with \\ non-uniform nodes \\ \(\dot{x}_{i}=A_{i}\mathbf{x}_{i},\ddot{x}_{i}=A_{i}^{2}\mathbf{x}_{i}\) \\ \end{tabular} &
\begin{tabular}{c} DQ finite element with \\ non-uniform nodes \\ \(\dot{u}_{i}=A_{i}\mathbf{u}_{i}\) for C\({}^{0}\) problems \\ \(\ddot{w}_{i}=\dot{A}_{i}^{2}\mathbf{w}_{i}\) for C\({}^{1}\) problems \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison among DQTFEM (Differential Quadrature Time Finite Element Method), DQTEM0 and the improved DQTEM. Before discussing the accuracy of the improved DQTEM, the definition of sympletic method is recalled first. If the Jacobi matrix \(\mathbf{J}\) of a method satisfies the following conditions  
\[\left\{ \begin{array}{l}\mathbf{J}_{1}\!\!=\!\!\mathbf{J}_{11}\mathbf{J}_{22}^{T}-\mathbf{J}_{12 }\mathbf{J}_{21}^{T}=\mathbf{I},\mathbf{J}_{2}=\mathbf{J}_{11}^{T}\mathbf{J}_{22}-\mathbf{J}_{21}^{T}\bm {J}_{12}=\mathbf{I}\\ \mathbf{J}_{3}=\mathbf{J}_{11}^{T}\mathbf{J}_{21}-\mathbf{J}_{2}^{T}\mathbf{J}_{11}=\mathbf{0},\mathbf{J}_{ 4}=\mathbf{J}_{12}^{T}\mathbf{J}_{22}-\mathbf{J}_{22}^{T}\mathbf{J}_{12}=\mathbf{0}\\ \mathbf{J}_{5}=\mathbf{J}_{11}\mathbf{J}_{12}^{T}-\mathbf{J}_{12}\mathbf{J}_{11}^{T}=\mathbf{0},\mathbf{J}_ {6}=\mathbf{J}_{21}\mathbf{J}_{22}^{T}-\mathbf{J}_{22}\mathbf{J}_{21}^{T}=\mathbf{0}\end{array} \right. \tag{38}\]  
then the method is symplectic which preserves energy and momentum of conservative system.