Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations

TL;DR

This paper demonstrates that certain symplectic Runge--Kutta methods applied to Hamiltonian PDEs can be embedded into a modified Hamiltonian flow with exponentially small error, ensuring near-conservation of energy for analytic initial data.

Contribution

It introduces a novel embedding technique for symplectic Runge--Kutta methods applied to Hamiltonian PDEs, achieving exponential accuracy in energy conservation.

Findings

Modified Hamiltonian is close to the original energy within $O(h^p)$

Exponential error bounds depend on the regularity and analyticity of initial data

Method applies to semilinear wave and nonlinear Schrödinger equations

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 $O(h^p)$-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 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.