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.

偏微分方程（PDEs）是科学和工程领域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，这些方程表示质量，能量和动量守恒。例如，飞机制造商对设计飞机机翼的效率、性能和安全性最大化很感兴趣。相关的偏微分方程为上述未知速度场和压力场的守恒定律。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求解偏微分方程。在现实的3D情况下，可能有数百万个未知数和方程。即使是最强大的计算机也无法用经典方法在合理的时间内解出这些方程。