# G. Tuba Masur

## "Exponential integrators for a finite dimensional Hamiltonian system and its relation to Klein-Gordon equations"

 Date: Thu, October 4, 2018 Time: 14:15 Place: Research I Seminar Room

Abstract: We modified a finite dimensional Hamiltonian system as a system whose fast and slow dynamics are predominantly separated. We numerically solved the modified system, a stiff system depending on a small parameter $$\varepsilon$$, through exponential Runge-Kutta integrators in recently proposed optimal balance algorithm. As a result, order reduction to $$O(1/\varepsilon)$$ disregarding order of exponential integrators is obtained for very small $$\varepsilon$$ values. To tackle with this problem, we want to build new exponential integrators uniform in $$\varepsilon$$. In this context, we have found a work describing uniformly accurate exponential integrator for Klein-Gordon equations promising, and to follow the same idea, we provided relation between the current system and described transformations in the Klein-Gordon equations. See [HO10] and [CM02] for exponential integrators, [GMO17] for optimal balance algorithm, [BFS18] for uniformly accurate exponential integrators, and [MO18] for relating the considered Hamiltonian system to the semilinear Klein-Gordon equation.

[BFS18] Simon Baumstark, Erwan Faou, and Katharina Schratz. Uniformly accurate exponential type integrators for Klein-Gordon equations with asymptotic convergence to the classical nls splitting. Mathematics of Computation, 87(311):1227-1254, 2018.

[CM02] Steven M Cox and Paul C Matthews. Exponential time differencing for stiff systems. Journal of Computational Physics, 176(2):430-455, 2002.

[GMO17] Georg A Gottwald, Haidar Mohamad, and Marcel Oliver. Optimal balance via adiabatic invariance of approximate slow manifolds. Multiscale Modeling and Simulation, 15(4):1404-1422, 2017.

[HO10] Marlis Hochbruck and Alexander Ostermann. Exponential integrators. Acta Numerica, 19:209-286, 2010.

[MO18] Haidar Mohamad and Marcel Oliver. $$H^s$$-class construction of an almost invariant slow subspace for the Klein-Gordon equation in the non-relativistic limit. Journal of Mathematical Physics, 59(5):051509, 2018.