Laura Noreña, University of Central Florida


Nearly Structure-Preserving Algorithms for Perturbations of Multi-Symplectic PDE's

Abstract: Given a differential equation that describes some behavior, it is desired that its numerical solution preserve as many properties of the exact solution as possible. For multi-symplectic partial differential equations (PDE's), numerical methods that preserve their structure have good energy and momentum preservation properties. Some examples of multi-symplectic equations are the non-linear Schroedinger, the Korteweg-de Vries (KdV), and the wave equations. Recently, these methods have been applied to PDE's with added weak dissipation; the results obtained show that when the damping terms are small the structure is preserved. Adding terms to a multi-symplectic PDE changes the structure, making it more suitable to model real-world problems. Intuitively, small changes in the equation lead to small changes in the multi-symplectic structure. The aim of this work is to generalize the multi-symplectic equation and determine how small the added terms should be, relative to the grid size, to preserve the structure up to the order of the method. To explore this we will be using well-known multi-symplectic integrators, such as the multi-symplectic Euler method, and we extend the idea of splitting methods for Hamiltonian systems to these equations. As a particular example, we will consider the KdV-Burger's equation and run numerical experiments to better understand the behavior of the method.

Work funded by the National Science Foundation

Collaborators: Brian Moore and Constance Schober (University Central Florida)