Theoretical Developments and Applications of Conservative Discretizations

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。随着系统状态的发展，这些量必须保持恒定，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，通常使用数值方法在长时间的计算机模拟中近似系统的状态。不幸的是，传统的数值方法不是保守的，因为它们不能保持不变量，这可能导致它们的近似值有很大的偏差。虽然存在一般的保守方法，但它们在长期模拟中可能表现出不稳定性，或者在具有多个不变量的大型动力系统中实现困难。一种新的保守方法被称为离散乘子法（DMM），它可以避免这些困难。本研究计划旨在探索两个平行的目标：1)DMM的理论发展，以及2)DMM的应用。在第一个目标中，将研究DMM理论的扩展。具体来说，我们将开发新的保守方法来求解时变偏微分方程。特别是，波动和色散现象是由这样的方程在空间和时间上建模的，它们以一种非平凡的方式相互作用，因此它们的不变量是保留的。我们将采用传统的空间近似数值方法和时间近似DMM方法，设计新的保守方法来研究这些时变问题。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DMM应用于分子动力学的长期模拟。虽然已经开发了专门的数值方法用于分子模拟，但它们不能保留能量，而能量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数据。这项工作将导致一种新的数值方法，对计算科学具有有利的长期性质，并将对物理科学和工程中的各种重要问题产生直接影响。

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.