|Date:||Thu, October 4, 2018|
|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.