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02/01/07 Colloquium
Dr. Brian Moore
University of Iowa
University of Iowa
The Beginnings of Backward Error Analysis for Multi-Symplectic Integration Methods
Abstract: Backward error analysis is the most important tool we have for showing that symplecticnumerical methods yield qualitatively accurate solutions for Hamiltonian systems of ODEsover long time intervals. The idea of backward error analysis is to find a system of differential equations, called the modified equations, which describe the solution behavior of the numerical method. For multi-symplecticmethods (algorithms that preserve the space/time symplecticstructure of a Hamiltonian PDE) the theory of backward error is only beginning to develop. The modified equations for these methods are constructed through asymptotic expansions, and one can provethat the modified equations always have a multi-symplecticstructure. In the case of linear systems, the expansions converge to the numerical scheme,showing that the modified equations are an exact representation of the method. Unfortunately, the expansions do not converge for nonlinear problems, but we are able to show that the numerical method reproduces the qualitative solution behavior of the original PDE to higher order accuracy. These ideas are also applied to dissipative PDEsyielding similar results.