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C. Wulff and M. Oliver,
Exponentially accurate Hamiltonian embeddings of symplectic A-stable
Runge-Kutta methods for Hamiltonian semilinear evolution equations,
P. Roy. Soc. Edinb. 146A (2016), 1265-1301.
Abstract:
We prove that a class of A-stable symplectic Runge-Kutta time
semidiscretizations (including the Gauss-Legendre methods) applied to
a class of semilinear Hamiltonian PDEs which are well-posed on spaces
of analytic functions with analytic initial data can be embedded into
a modified Hamiltonian flow up to an exponentially small error. As a
consequence, such time-semidiscretizations conserve the modified
Hamiltonian up to an exponentially small error. The modified
Hamiltonian is O(hp)-close to the original energy where p is the
order of the method and h the time step-size. Examples of such
systems are the semilinear wave equation or the nonlinear
Schrödinger equation with analytic nonlinearity and periodic
boundary conditions. Standard Hamiltonian interpolation results do
not apply here because of the occurrence of unbounded operators in the
construction of the modified vector field. This loss of regularity in
the construction can be taken care of by projecting the PDE to a
subspace where the operators occurring in the evolution equation are
bounded and by coupling the number of excited modes as well as the
number of terms in the expansion of the modified vector field with the
step size. This way we obtain exponential estimates of the form
O(exp(-c/h1/(1+q))) with
c>0 and q≥0; for the
semilinear wave equation, q=1, and for the nonlinear Schrödinger
equation, q=2. We give an example which shows that analyticity of
the initial data is necessary to obtain exponential estimates.
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