In this paper, we show that the eigenvalues and eigenvectors of the spectral discretisation matrices resulted from the Legendre dual-Petrov-Galerkin (LDPG) method for the $m$th-order initial value problem (IVP): $u^{(m)}(t)=\sigma u(t),\, t\in (-1,1)$ with constant $\sigma\not=0$ and usual initial conditions at $t=-1,$ are associated with the generalised Bessel polynomials (GBPs). The essential idea of the analysis is to properly construct the basis functions for the solution and its dual spaces so that the matrix of the $m$th derivative is an identity matrix, and the mass matrix is then identical or approximately equals to the Jacobi matrix of the three-term recurrence of GBPs with specific integer parameters. This allows us to characterise the eigenvalue distributions and identify the eigenvectors. As a by-product, we are able to answer some open questions related to the very limited known results on the collocation method at Legendre points (studied in 1980s) for the first-order IVP, by reformulating it into a Petrov-Galerkin formulation. Moreover, we present two stable algorithms for computing zeros of the GBPs, and develop a general space-time spectral method for evolutionary PDEs using either the matrix diagonalisation, which is restricted to a small number of unknowns in time due to the ill-conditioning but is fully parallel, or the QZ decomposition which is numerically stable for a large number of unknowns in time but involves sequential computations. We provide ample numerical results to demonstrate the high accuracy and robustness of the space-time spectral methods for some interesting examples of linear and nonlinear wave problems.
翻译:在本文中,我们显示,光谱离散矩阵的egen值和偏差值与光谱离散矩阵的常规条件$gm=0美元和通常初始条件$t=1美元相关。 分析的基本想法是适当构建解决方案及其双空的基函数,以使美元序列初始值问题(IVP):$u ⁇ (m)}(t) ⁇ sgma u(t),\\\gma\\(t),\\\\(t)\(1,1)美元=0美元,与普通的贝塞尔多语系(GBPPs)相联。正确构建了解决方案及其双空空间基值的基值函数,使美元序列的基数稳定值函数是一个身份矩阵矩阵模型,然后质量矩阵与含有特定整数参数的三期复发数相同或大致相等。 这使得我们能够描述易变值分布,但是作为产品,我们能够解解一些与已知的直径直径直径直的直径直数相关的问题, 在1980年的直径直径直径直径直径直的直径直的直的直径直径直径直数值结果中, 在图中, 解方法中显示的直径直到直径直径直地算法的直地算法的直地算法的直地算算法的直到直地, 直径直地方法显示的直到正向方向法系的直径直径向方向法的直的直到直到直到直到方向法制成。