Linearly implicit exponential integrators for damped Hamiltonian PDEs

Structure-preserving linearly implicit exponential integrators are constructed for Hamiltonian partial differential equations with linear constant damping. Linearly implicit integrators are derived by polarizing the polynomial terms of the Hamiltonian function and portioning out the nonlinearly of consecutive time steps. They require only a solution of one linear system at each time step. Therefore they are computationally more advantageous than implicit integrators. We also construct an exponential version of the well-known one-step Kahan's method by polarizing the quadratic vector field. These integrators are applied to one-dimensional damped Burger's, Korteweg-de-Vries, and nonlinear Schr\"odinger equations. Preservation of the dissipation rate of linear and quadratic conformal invariants and the Hamiltonian is illustrated by numerical experiments

Structure-preserving Finite Difference Methods for Linearly Damped Differential Equations

Differential equations (DEs) model a variety of physical phenomena in science and engineering. Many physical phenomena involve conservative or dissipative forces, which manifest themselves as qualitative properties of DEs that govern these phenomena. Since only a few and simplistic models are known to have exact solutions, approximate solution techniques, such as numerical integration, are used to reveal important insights about solution behavior and properties of these models. Numerical integrators generally result in undesirable quantitative and qualitative errors . Standard numerical integrators aim to reduce quantitative errors, whereas geometric (numerical) integrators aim to reduce or eliminate qualitative errors, as well, in order to improve the accuracy of numerical solutions. It is now widely recognized that geometric (or structure-preserving) integrators are advantageous compared to non-geometric integrators for DEs, especially for long time integration. Geometric integrators for conservative DEs have been proposed, analyzed, and investigated extensively in the literature. The motif of this thesis is to extend the idea of structure preservation to linearly damped DEs. More specifically, we develop, analyze, and implement geometric integrators for linearly damped ordinary and partial differential equations (ODEs and PDEs) that possess conformal invariants, which are qualitative properties that decay exponentially along any solution of the DE as the system evolves over time. In particular, we derive restrictions on the coefficient functions of exponential Runge-Kutta (ERK) numerical methods for preservation of certain conformal invariants of linearly damped ODEs. An important class of these methods is shown to preserve the damping rate of solutions of damped linear ODEs. Linearly stability and order of accuracy for some specific cases of ERK methods are investigated. Geometric integrators for PDEs are designed using structure-preserving ERK methods in space, time, or both. These integrators for PDEs are also shown to preserve additional structure in certain special cases. Numerical experiments illustrate higher order accuracy and structure preservation properties of various ERK based methods, demonstrating clear advantages over non-structure-preserving methods, as well as usefulness for solving a wide range of DEs

Comparison of Second Order Conformal Symplectic Schemes with Linear Stability Analysis

Numerical methods for solving linearly damped Hamiltonian ordinary differential equations are analyzed and compared. The methods are constructed from the well-known Störmer-Verlet and implicit midpoint methods. The structure preservation properties of each method are shown analytically and numerically. Each method is shown to preserve a symplectic form up to a constant and are therefore conformal symplectic integrators, with each method shown to accurately preserve the rate of momentum dissipation. An analytical linear stability analysis is completed for each method, establishing thresholds between the value of the damping coefficient and the step-size that ensure stability. The methods are all second order and the preservation of the rate of energy dissipation is compared to that of a third order Runge-Kutta method that does not preserve conformal properties. Numerical experiments will include the damped harmonic oscillator and the damped nonlinear pendulum

Geometric Numerical Integration

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods

Physical Significance of Noether Symmetries

In this paper, we will trace the development of the use of symmetry in discussing the theory of motion initiated by Emmy Noether in 1918. Though it started with its use in classical mechanics, and has been heavily used in engineering applications of mechanics, it came into its own in relativity, and quantum theory and their applications in particle physics and field theory. It will be beyond the scope of this article to explain the quantum field theory applications in any detail, but the base for understanding it will be provided here. We will also go on to discuss an insight from some more mathematical developments related to Noether symmetry