Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations
This provides rigorous exponentially accurate long-time energy conservation for symplectic integrators applied to a class of Hamiltonian PDEs, addressing a gap where standard backward error analysis fails due to unbounded operators.
The authors prove that A-stable symplectic Runge-Kutta methods for semilinear Hamiltonian PDEs with analytic initial data can be embedded into a modified Hamiltonian flow with exponentially small error, leading to near-conservation of a modified energy. For the semilinear wave equation, the error is O(exp(-c/h^{1/2})), and for the nonlinear Schrödinger equation, O(exp(-c/h^{1/3})).
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 $O(h^p)$-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 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 $O(\exp(-c/h^{1/(1+q)}))$ with $c>0$ and $q \geq 0$; 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.