Theoretical Developments and Applications of Conservative Discretizations

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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.