Modern numerical methods for partial differential equations

偏微分方程的现代数值方法

• 批准号: RGPIN-2016-05983
• 负责人: Lui, ShiuHong(Shaun)
• 金额: $ 1.09万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=709835
• 关键词: Modern; numerical; methods; partial; differential

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.

部分微分方程（PDE）是科学和工程学研究中最重要的对象之一。这些方程的解决方案将揭示正在研究的系统的特性。这些方程经常表达质量，能量和动量的保护。例如，飞机制造商有兴趣设计飞机的机翼，使效率，性能和安全性最大化。相关的PDE是上述未知速度和压力场的保护定律。不幸的是，这些方程式的明确解决方案非常罕见。在一般的科学家和工程师中，依靠数值方法来解决PDE。在现实的3D案例中，可能有数百万个未知数和方程式。最强大的计算机无法使用经典方法在合理时间内求解这些方程。 光谱方法是求解PDE的经典数值方法。如果PDE独立于时间并且解决方案是平滑的，则光谱方法呈指数收敛，这意味着，对于相同的准确性顺序，与数百万未知数的未知数相比，仅需要数千个未知数，而其他未指数收敛的方法则需要。因此，与其他方法相比，光谱方法可以更快地获得解决方案。不幸的是，如果PDE取决于时间，则经典光谱方法不会成倍收敛。时空光谱方法是新方法，它们确实会呈指数融合，并且仅在过去十年内出现。 在当前的发现周期中，我已证明了用于热方程的时空光谱法的指数收敛，这是描述人体温度分布的时间依赖性PDE。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PDE。之后，可以考虑非线性方程。另一个重要方面是为在时空光谱方法中出现的方程式设计快速求解器。最后，经典光谱方法仅适用于矩形几何形状。对于复杂几何形状上的问题，可以考虑光谱元素方法（经典光谱离散化的概括）。 要探索的另一个主题涉及分数PDE的数值方法。经典的PDE模型局部现象，而分数PDE用于表现出非局部行为的系统。分数PDE主要在过去十年中进行了研究，并且是数学最活跃的领域之一。分数PDE的数值方法仅在过去五年中出现。 最近，我对有限差异方法进行了1d差异分数方程的收敛分析。该程序的某些目的包括将分析扩展到更高的维度和依赖时间的问题，以及为所得方程设计现代快速求解器。