Mats Vermeeren
It is well-known that if a symplectic integrator is applied to a Hamiltonian system, then the modified equation, whose solutions interpolate the numerical solutions, is again Hamiltonian. We investigate this property from the variational side. We present a technique to construct a Lagrangian for the modified equation from the discrete Lagrangian of a variational integrator.
Mats Vermeeren
Discretizations of differential equations are often studied through their modified equation. This is a differential equation, usually obtained as a power series, with solutions that exactly interpolate the discretization. By comparing the Störmer-Verlet discretization of the harmonic oscillator with its modified equation, we obtain a relatively simple derivation of the expansion \[ \left( \arcsin \frac{h}{2} \right)^2 = \frac{1}{2} \sum_{k=1}^\infty \frac{(k-1)!^2}{(2k)!} h^{2k}, \] which can be used to show that $ζ(2) = \frac{π^2}{6}$.
Mats Vermeeren
Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation describing parasitic oscillations. We observe that a Lagrangian for the principal modified equation can be constructed using the same technique as in the case of non-degenerate Lagrangians. Furthermore, we construct the full system of modified equations by doubling the dimension of the discrete system in such a way that the principal modified equation of the extended system coincides with the full system of modified equations of the original system. We show that the extended discrete system is Lagrangian, which leads to a construction of a Lagrangian for the full system of modified equations.
Mats Vermeeren
Kepler's first law states that the orbit of a point mass with negative energy in a classical gravitational potential is an ellipse with one of its foci at the gravitational center. In numerical simulations of this system one often observes a slight precession of the ellipse around the gravitational center. Using the Lagrangian structure of modified equations and a perturbative version of Noether's theorem, we provide leading order estimates of this precession for the implicit MidPoint rule (MP) and the Störmer-Verlet method (SV). Based on those estimates we construct some new numerical integrators that perform significantly better than MP and SV on the Kepler problem.