On an asymptotic method for computing the modified energy for symplectic methods
This provides a more accurate and efficient method for computing shadow energies in Hamiltonian simulations, which is relevant for numerical analysts working on long-time integration.
The authors improved an algorithm for computing the modified energy in symplectic discretizations of Hamiltonian systems, achieving arbitrary high order accuracy via Richardson extrapolation and demonstrating exponentially small drift. Numerical examples confirm the theory.
We revisit an algorithm by Skeel et al. for computing the modified, or shadow, energy associated with the symplectic discretization of Hamiltonian systems. By rephrasing the algorithm as a Richardson extrapolation scheme arbitrary high order of accuracy is obtained, and provided error estimates show that it does capture the theoretical exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory.