Spectral methods solve elliptic partial differential equations (PDEs) numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Spectral methods that converge spectrally in both space and time have appeared recently. This paper is a continuation of the authors' previous works on Legendre and Chebyshev space-time methods for the heat equation. Here space-time spectral collocation methods for the Schrodinger, wave, Airy and beam equations are proposed and analyzed. In particular, a condition number estimate of each global Chebyshev space-time operator is shown. The analysis requires new estimates of eigenvalues of some spectral derivative matrices. In particular, it is shown that the real part of every eigenvalue of the third-order Chebyshev derivative matrix is positive and bounded away from zero, settling a twenty-year-old conjecture. Similarly, the real part of every eigenvalue of the fourth-order Chebyshev derivative matrix with Dirichlet boundary conditions is shown to be also positive and bounded away from zero. Numerical results verify the theoretical results, and demonstrate that the space-time methods also work well for some common nonlinear PDEs.

当解析时，频谱方法用数值方法求解椭圆偏微分方程（PDE），其误差由模式数的指数衰减函数限制。对于与时间有关的问题，几乎所有焦点都集中在时间导数的低阶有限差分方案和空间导数的频谱方案上。最近出现了在空间和时间上都在光谱上收敛的光谱方法。本文是作者先前关于Legendre和Chebyshev时空方法的热方程的延续。提出并分析了薛定inger方程，波动方程，艾里方程和射束方程的时空频谱配置方法。特别地，示出了每个全局切比雪夫时空算子的条件数估计。该分析需要对某些频谱导数矩阵的特征值进行新的估计。特别地，表明三阶切比雪夫导数矩阵的每个特征值的实部为正，并且远离零，从而解决了二十年前的猜想。同样，具有Dirichlet边界条件的四阶Chebyshev导数矩阵的每个特征值的实部也显示为正，并且有界于零。数值结果验证了理论结果，并证明了时空方法对于某些常见的非线性PDE也适用。具有Dirichlet边界条件的四阶Chebyshev导数矩阵的每个特征值的实部也显示为正，并且定界为零。数值结果验证了理论结果，并证明了时空方法对于某些常见的非线性PDE也适用。具有Dirichlet边界条件的四阶Chebyshev导数矩阵的每个特征值的实部也显示为正，并且定界为零。数值结果验证了理论结果，并证明了时空方法对于某些常见的非线性PDE也适用。