Theoretical Developments and Applications of Conservative Discretizations

保守离散化的理论发展与应用

• 批准号: RGPIN-2019-07286
• 负责人: Wan, Andy
• 金额: $ 1.17万
• 依托单位: University of Northern British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759161
• 关键词: Theoretical; Developments; Applications; Conservative; Discretizations

Many important dynamical systems in physical sciences and engineering possess important geometric structures, such as invariant quantities. Such quantities must stay constant as the state of the system evolves and they are essential for understanding the long-term behaviour of these systems. In order to study these complex dynamical systems, numerical methods are often used to approximate the state of the system on computer simulations over long periods of time. Unfortunately, traditional numerical methods are not conservative, as they do not preserve invariants which can lead to large deviations in their approximations. While general conservative methods exist, they can exhibit instabilities over long-term simulations or have difficulties with implementation for large dynamical systems with multiple invariants. A new class of conservative methods known as the Discrete Multiplier Method (DMM) was recently developed which can avoid these difficulties. This research program seeks to explore two parallel objectives: 1) Theoretical developments of DMM, and 2) Applications of DMM. In the first objective, extensions to the theory of DMM will be investigated. Specifically, we will develop new conservative methods for time-dependent partial differential equations. In particular, wave and dispersive phenomena are modelled by such equations in both space and time and they interact in a nontrivial way so that their invariants are preserved. By applying traditional numerical methods in spatial approximation and DMM in temporal approximation, we will devise new conservative methods to study these time-dependent problems. Moreover, we will study the connections of DMM with existing general conservative methods and compare their stability properties. In the second objective, we will explore novel applications of DMM in physical sciences and engineering. Specifically, we intend to develop extensions of DMM to simulate flows on manifolds. Such flows are important in many areas as they appear in classical mechanics, control theory, image processing, neural networks and optimizations. Using the DMM approach, new conservative methods will be developed to study this large class of important problems. In addition, we will apply DMM to long-term simulations in molecular dynamics. While specialized numerical methods have been developed for molecular simulations, they do not preserve the energy, which is an important invariant of these systems. Instead, we will devise new conservative methods for molecular dynamics and compare the statistics of different molecular models. This work will lead to a new class of numerical methods with favourable long-term properties for computational science and will have a direct impact on a variety of important problems from physical sciences and engineering.

在物理科学和工程中，许多重要的动力系统都具有重要的几何结构，如不变量。随着系统状态的演变，这些量必须保持不变，它们对于理解这些系统的长期行为至关重要。 为了研究这些复杂的动力系统，经常使用数值方法来近似系统在长时间内的计算机模拟状态。不幸的是，传统的数值方法是不保守的，因为它们不保留不变量，这可能导致其近似值的大偏差。虽然存在一般的保守方法，但它们可能在长期模拟中表现出不稳定性，或者难以实现具有多个不变量的大型动态系统。 最近发展了一类新的保守方法，称为离散乘子法（DMM），可以避免这些困难。本研究计划旨在探索两个平行的目标：1）DMM的理论发展，和2）DMM的应用。在第一个目标中，扩展的DMM理论将进行调查。具体来说，我们将开发新的保守方法的时间依赖的偏微分方程。特别是，波和色散现象是由这样的方程在空间和时间建模，它们以一种非平凡的方式相互作用，使它们的不变量被保存。通过在空间近似中应用传统的数值方法，在时间近似中应用DMM，我们将设计新的保守方法来研究这些时间依赖的问题。此外，我们将研究DMM与现有的一般保守方法的连接，并比较它们的稳定性。在第二个目标中，我们将探索DMM在物理科学和工程中的新应用。具体来说，我们打算开发DMM的扩展来模拟流形上的流动。这种流动在许多领域都很重要，因为它们出现在经典力学、控制理论、图像处理、神经网络和优化中。使用DMM的方法，新的保守的方法将被开发来研究这一大类重要的问题。此外，我们将DMM应用于分子动力学的长期模拟。虽然专门的数值方法已被开发用于分子模拟，但它们不保持能量，这是这些系统的重要不变量。相反，我们将为分子动力学设计新的保守方法，并比较不同分子模型的统计。这项工作将导致一类新的数值计算方法，具有良好的长期性能的计算科学，并将产生直接影响的各种重要问题，从物理科学和工程。