Abstract
A useful method for understanding discretization error in the numerical solution of ODEs is to compare the system of ODEs with the modified equations, the equations solved by the numerical solution, which are obtained through backward error analysis. Using symplectic integration for Hamiltonian ODEs provides more insight into the modified equations. In this thesis, the ideas of symplectic integration are extended to Hamiltonian PDEs, such that the symplectic structure in both space and time is exactly preserved. This paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations. In particular, the multi-symplectic Euler, explicit mid-point, and Preissman box schemes are considered for general multi-symplectic equations. It is shown that these methods exactly preserve a multi-symplectic conservation law, as well as semi-discrete conservation laws of energy and momentum, and in some specific cases other fully discrete conservation laws. For a full discretization, local conservation laws of energy and momentum are not, in general, preserved exactly, but using Taylor series expansions one obtains a modified multi-symplectic PDE. Then, the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution. It is also shown that the modified equations for linear problems converge to the numerical scheme, and numerical dispersion relations are also derived, giving more insight into the behavior of each method. The idea of multi-symplectic integration and modified equations are also applied to Hamiltonian PDEs with added dissipation. It is shown that it is possible to numerically preserve dissipation properties of the PDE, making it clear that a key characteristic of multi-symplectic integrators is that there is no dissipation added by the discretization. Various model problems are considered through out the thesis, including the Korteweg-de Vries and nonlinear wave equations.