Exponentially accurate Hamiltonian embeddings of symplectic Astable RungeKutta methods for Hamiltonian semilinear evolution equations
Wulff, Claudia and Oliver, M (2016) Exponentially accurate Hamiltonian embeddings of symplectic Astable RungeKutta methods for Hamiltonian semilinear evolution equations PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION AMATHEMATICS, 146 (6). pp. 12651301.

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Abstract
We prove that a class of Astable symplectic RungeKutta time semidiscretizations (including the GaussLegendre methods) applied to a class of semilinear Hamiltonian PDEs which are wellposed 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 timesemidiscretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(h^p)close to the original energy where p is the order of the method and h the time stepsize. Examples of such systems are the semilinear wave equation or the nonlinear Schr\"odinger 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/h^{1/(1+q)})) with c>0 and q \geq 0; for the semilinear wave equation, q=1, and for the nonlinear Schr\"odinger equation, q=2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.
Item Type:  Article  

Divisions :  Faculty of Engineering and Physical Sciences > Mathematics  
Authors : 


Date :  1 December 2016  
DOI :  :10.1017/S0308210515000852  
Copyright Disclaimer :  © Royal Society of Edinburgh 2016  
Related URLs :  
Depositing User :  Symplectic Elements  
Date Deposited :  28 Mar 2017 13:10  
Last Modified :  05 Jul 2019 16:16  
URI:  http://epubs.surrey.ac.uk/id/eprint/805291 
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