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.

物理科学和工程中许多重要的动力系统具有重要的几何结构，例如不变量。随着系统状态的发展，这种数量必须保持稳定，并且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态系统，数值方法通常用于长时间的计算机模拟近似系统状态。不幸的是，传统的数值方法不是保守的，因为它们不能保留不变的，从而导致其近似值较大。尽管存在一般的保守方法，但它们可以在长期模拟上表现出不稳定性，也可以在具有多个不变性的大型动力系统实施方面遇到困难。 最近开发了一种新的称为离散乘数方法（DMM）的保守方法，可以避免这些困难。该研究计划旨在探索两个并行目标：1）DMM的理论发展，以及2）DMM的应用。在第一个目标中，将研究对DMM理论的扩展。具体而言，我们将开发新的保守方法，以依赖时间依赖时间的部分微分方程。特别是，波浪和色散现象是通过时空和时间上的这些方程建模的，它们以非平凡的方式进行相互作用，以便保留其不变性。通过在时间近似中应用传统的数值方法在空间近似和DMM中，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联系，并比较其稳定性。在第二个目标中，我们将探索DMM在物理科学和工程中的新颖应用。具体而言，我们打算开发DMM的扩展，以模拟歧管上的流。这些流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方法，将开发新的保守方法来研究这一大类重要问题。此外，我们将在分子动力学中将DMM应用于长期模拟。尽管已经开发了用于分子模拟的专门数值方法，但它们并不能保留能量，这是这些系统的重要不变。取而代之的是，我们将设计新的保守方法来用于分子动力学，并比较不同分子模型的统计数据。这项工作将导致一类新的数值方法具有有利的计算科学长期特性，并将直接影响物理科学和工程的各种重要问题。