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• W1887180641 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 conserve a modified energy up to an exponentially small error. This modified energy is O(h^p)-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 Schrodinger equation with analytic nonlinearity and periodic boundary conditions. Standard backward error analysis does 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 vectorfield with the stepsize. This way we obtain exponential estimates of the form O(exp(-beta/h^{1/(1+q)})) with beta>0 and q geq 0; for the semilinear wave equation, q=1, and for the nonlinear Schrodinger equation, q=2." @default.
• W1887180641 created "2016-06-24" @default.
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• W1887180641 date "2012-08-22" @default.
• W1887180641 modified "2023-10-01" @default.
• W1887180641 title "Approximate energy conservation of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations" @default.
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