Space-time Spectral Methods for Differential equations

数学，特别是微分方程，是科学和工程的语言。用偏微分方程（PDE）解释了动脉中血液和机翼上空气的流动。不幸的是，这些方程的精确解很少。科学家和工程师依靠数值方法来求解这些方程。经典技术可能会导致数百万个非线性方程的求解，即使是最强大的计算机也会负担沉重。谱方法解决时间无关的偏微分方程数值误差由一个指数衰减函数的模式的数量时，解决方案是光滑的。它们所需的未知数比其他方法少得多，可以达到相当的精度。我和我的学生一直在研究一类新的谱方法，用于时间依赖的偏微分方程，称为时空谱方法，它在谱上收敛（即，指数地）在空间和时间上对于重要的线性偏微分方程。我们严格证明了这些方法的谱收敛性和条件数估计。后者量化了问题求解的难易程度，从数值分析的角度来看，是最重要的研究对象。我们还数值证明了时空谱方法对许多类非线性偏微分方程是有效的，包括一些最重要的偏微分方程的应用。 目前，空时谱方法最大的缺点是必须同时求解所有时间上的所有未知量。这对3D问题和/或非线性问题提出了严重的困难，并且是本提案中将解决的主要问题。我们提出了新的算法，可以解决并行计算机上的偏微分方程，以加快计算速度。这个建议的第二个目标是线性偏微分方程的时空超球谱方法。超球面谱方法是近年来发展起来的一类谱方法，它比传统的谱方法具有更低的存储需求和更高的稳定性。它是数值偏微分方程最有前途的方法之一。本研究的另一个目的是对延迟微分方程的时空谱方法进行研究。最后的目标是进一步的进展，一个25岁的开放问题的空间-时间谱四阶导数算子的频谱的属性。空时谱方法是求解线性和非线性偏微分方程的鲁棒和通用的方法。他们已被证明是有效的许多类型的偏微分方程拥有不同的数学性质。我们相信，在加速时空谱方法求解过程方面的任何进展都可能对科学计算产生重大影响，并将为科学家和工程师发现新现象和创新提供工具。最后，本研究将为HQP提供有价值的理论和计算训练。

Mathematics, in particular, the area known as differential equations, is the language of science and engineering. The flow of blood in arteries and air over aircraft wings are explained by partial differential equations (PDEs). Unfortunately, exact solutions of these equations are rare. Scientists and engineers rely on numerical methods to solve these equations. Classical techniques can result in many millions of nonlinear equations to solve, taxing even the most powerful computers. Spectral methods solve time-independent PDEs numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is smooth. They require far fewer unknowns than other methods achieving comparable accuracy. My students and I have been studying a new class of spectral methods for time-dependent PDEs, called space-time spectral methods, that converge spectrally (i.e., exponentially) in both space and time for important linear PDEs. We have shown rigorously the spectral convergence, as well as condition number estimates of these methods. The latter quantifies the degree of difficulty of solving the problems, and are the most important objects of study from the point of view of numerical analysis. We have also demonstrated numerically that space-time spectral methods are effective for many classes of nonlinear PDEs, including some of the most important PDEs in applications. Currently, the most serious drawback of space-time spectral method is that all unknowns over all times must be solved simultaneously. This presents a serious difficulty for 3D problems and/or nonlinear problems, and is the main issue that will be addressed in this proposal. We propose new algorithms that can solve the PDEs on parallel computers to speed up the computations. A second objective of this proposal is a space-time ultra-spherical spectral method for linear PDEs. The ultra-spherical spectral method is a recent class of spectral methods that possesses lower memory requirement and is more stable than traditional spectral methods. It is one of the most promising methods for numerical PDEs. Another objective of this research is a space-time spectral method for delay differential equations which are used extensively to model population dynamics. The final objective is further progress on a 25-year old open problem on a property about the spectrum of a space-time spectral fourth derivative operator. Space-time spectral methods are robust and universal solvers for linear and nonlinear PDEs. They have been shown to be effective for many types of PDEs possessing different mathematical properties. We believe that any progress in accelerating the solution process of space-time spectral methods can have a significant impact on scientific computing, and will provide tools to assist scientists and engineers in their discovery of new phenomena and innovations. Finally this research will provide valuable theoretical and computational training for HQPs.