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.
Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Bhatt, Ashish, "Structure-preserving Finite Difference Methods for Linearly Damped Differential Equations" (2016). Electronic Theses and Dissertations. 5630.