Modern numerical methods for partial differential equations

偏微分方程是科学和工程中最重要的研究对象之一。这些方程的解将揭示所研究系统的性质。通常，这些方程表达了质量，能量和动量的守恒。例如，飞机制造商有兴趣设计飞机机翼，使效率、性能和安全性最大化。相关的偏微分方程是上面提到的未知速度场和压力场的守恒律。不幸的是，这些方程的显式解非常罕见。一般来说，科学家和工程师依靠数值方法来解决偏微分方程。在现实的3D情况下，可能有数百万个未知数和方程。最强大的计算机无法使用经典方法在合理的时间内解决这些方程。谱方法是求解偏微分方程的经典数值方法。如果偏微分方程是时间无关的，并且解是光滑的，那么谱方法是指数收敛的，这意味着对于相同的精度，只需要数千个未知数，而其他方法则需要数百万个未知数。因此，谱方法可以得到的解决方案相比，其他方法快得多。不幸的是，如果偏微分方程是时间相关的，经典的谱方法不收敛指数。空时谱方法是一种指数收敛的新方法，仅在过去十年中出现。在当前的发现周期中，我已经证明了热方程的时空谱方法的指数收敛性，热方程是一个具有重要意义的时间相关PDE，它描述了物体的温度分布。与我的学生一起，下一步是重复科学和工程中发生的其他标准线性偏微分方程。之后，可以考虑非线性方程。另一个重要的方面是设计快速求解方程中出现的时空谱方法。最后，经典谱方法只适用于矩形几何。对于复杂几何问题，可以考虑谱元法，这是经典谱离散化的推广。另一个要探讨的主题涉及分数偏微分方程的数值方法。经典偏微分方程模型局部现象，而分数偏微分方程是表现出非局部行为的系统。分数阶偏微分方程是近十年来研究最多的数学问题，也是数学中最活跃的领域之一。分数阶偏微分方程的数值方法是在过去五年中才出现的。最近，我对一维时间无关分数阶方程的有限差分方法进行了收敛性分析。该计划的一些目标包括将分析扩展到高维和时间相关问题，并为所得方程设计现代快速求解器。

Partial differential equations (PDEs) are among the most important objects of study in science and engineering. Solution of these equations would reveal the properties of the system under study. Very often, these equations express conservations of mass, energy and momenta. For instance, aircraft manufacturers are interested in designing wings of aircraft maximizing efficiency, performance and safety. The relevant PDEs are the conservation laws mentioned above for the unknown velocity and pressure fields. Unfortunately, explicit solutions of these equations are very rare. In general scientists and engineers rely on numerical methods to solve PDEs. In realistic 3D cases, there may be many millions of unknowns and equations. The most powerful computers are not able to solve these equations in reasonable time using classical methods.***Spectral methods are classical numerical methods to solve PDEs. If the PDE is time independent and the solution is smooth, then spectral methods converge exponentially, meaning that for the same order of accuracy, only thousands of unknowns are required compared to many millions of unknowns for other methods which don't converge exponentially. Hence spectral methods can obtain the solution much quicker compared to other methods. Unfortunately, if the PDE is time dependent, the classical spectral method does not converge exponentially. Space-time spectral methods are new methods which do converge exponentially and have appeared only within the past decade.***In the current Discovery cycle, I have proven exponential convergence of a space-time spectral method for the heat equation, a time dependent PDE of great significance which describes the temperature distribution of a body. Together with my students, the next step is to repeat for other standard linear PDEs occurring in science and engineering. After that, nonlinear equations can be considered. Another important aspect is to design fast solvers for the equations which arise in the space-time spectral method. Finally, the classical spectral method works only for rectangular geometry. For problems on complex geometry, the spectral element method, a generalization of the classical spectral discretization, can be considered.***Another topic to be explored involves numerical methods for fractional PDEs. Classical PDEs model local phenomena, while fractional PDEs are for systems exhibiting nonlocal behaviour. Fractional PDEs have been studied mostly in the past decade and have been one of the most active areas of mathematics. Numerical methods for fractional PDEs have only appeared in the past five years.***Very recently, I have performed a convergence analysis of a finite difference method for a 1D time independent fractional equation. Some of the aims of this program include extending the analysis to higher dimensions and time dependent problems, and designing modern fast solvers for the resultant equations.**