Justification of a variational construction of approximate slow manifolds for Hamiltonian two-scale systems with strong gyroscopic forces

• Applicant: Professor Dr. Marcel Oliver
• Subject Area: Mathematics
• Term: from 2015 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 278124199

• Final Report Year: 2022

Final Report Abstract

The project was aiming at a characterization of slow manifolds in two-scale systems with strong gyroscopic forces in finite and infinite dimensional dynamical systems. For the infinitedimensional case, we focused on the semilinear Klein–Gordon equation in the nonrelativistic limit as a prototype example. In the finite-dimensional case, we were able to prove several new results that extend the existing theory in the field, which is generally mature and even classical. In the infinite-dimensional case, we developed new methods to characterize slow manifolds and were able to justify this construction rigorously over finite intervals of time. The full justification of an alternative variational construction remains open. We encountered substantial technical obstacles that put into question whether our initial conjectures are actually true. As a result, we spent a substantial amount of time in the second project phase to work on numerical schemes that allow to either verify or falsify these conjectures by simulation. To do so, it is essential to use problem-adapted numerical schemes as a conventional numerical scheme would require unreasonably short time-steps as the separation of slow and fast time scale increases. A number of such methods already exist in the literature, each specifically adapted to a very specific special case. Our contribution is that we were able to develop a new approach for highly oscillatory quadrature. This method can be used as part of a time-integrator for general dynamical systems with a single fast frequency and does not depend on problem-specific precomputations. In summary, even though we were not able to solve the initial problem fully, we successfully developed new analytical and numerical tools which bring us within reach of the initial goal.

Publications

• Comparison of variational balance models for the rotating shallow water equations, J. Fluid Mech. 822 (2017), 689–716
  D.G. Dritschel, G.A. Gottwald, and M. Oliver
  
• Optimal balance via adiabatic invariance of approximate slow manifolds, Multiscale Model. Simul. 15 (2017), 1404–1422
  G.A. Gottwald, H. Mohamad, and M. Oliver
  
• Hs-class construction of an almost invariant slow subspace for the Klein–Gordon equation in the non-relativistic limit, J. Math. Phys., 59 (2018), 051509
  H. Mohamad and M. Oliver
  
• A direct construction of a slow manifold for a semilinear wave equation of Klein-Gordon type, J. Differential Equations 267 (2019), 1–14
  H. Mohamad and M. Oliver
  
• High-order uniformly accurate time integrators for semilinear wave equations of Klein-Gordon type in the non-relativistic limit
  H. Mohamad and M. Oliver
  
• H. Mohamad and M. Oliver, Numerical integration of functions of a rapidly rotating phase, SIAM J. Num. Anal. 59 (2021), 2310–2319
  H. Mohamad and M. Oliver
  
• Energy asymptotics for the strongly damped Klein-Gordon equation
  H. Mohamad
  
• Quasi-convergence of an implementation of optimal balance by backward-forward nudging
  G.T. Masur, H. Mohamad and M. Oliver,