Approximate energy conservation of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations

摘要: We prove that a class of A-stable symplectic Runge--Kutta time semidiscretizations (including the Gauss--Legendre methods) applied to semilinear Hamiltonian PDEs which are well-posed on spaces analytic functions with initial data conserve modified energy up an exponentially small error. This is O(h^p)-close original where p order method and h step-size. Examples such systems wave equation or nonlinear Schr\"odinger nonlinearity periodic boundary conditions. Standard backward error analysis does not apply here because occurrence unbounded operators in construction vector field. loss regularity can be taken care by projecting PDE subspace occurring evolution bounded coupling number excited modes as well terms expansion vectorfield stepsize. way we obtain exponential estimates form O(\exp(-\beta/h^{1/(1+q)})) \beta>0 q \geq 0; for equation, q=1, q=2.