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M. Oliver and S. Vasylkevych,
A new construction of modified equations for
variational integrators,
submitted for publication.
Abstract:
The construction of modified equations is an important step in the
backward error analysis of symplectic integrator for Hamiltonian
systems. In the context of partial differential equations, the
standard construction leads to modified equations with increasingly
high frequencies which increase the regularity requirements on the
analysis. In this paper, we consider the next order modified equations
for the implicit midpoint rule applied to the semilinear wave equation
to give a proof-of-concept of a new construction which works directly
with the variational principle. We show that a carefully chosen
change of coordinates yields a modified system which inherits its
analytical properties from the original wave equation. Our method
systematically exploits additional degrees of freedom by modifying the
symplectic structure and the Hamiltonian together.
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