Line Integral Solution of Differential Problems

In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used. The methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references

Spectrally accurate space-time solution of Hamiltonian PDEs

Recently, the numerical solution of multi-frequency, highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. When the problem derives from the space semi- discretization of (possibly Hamiltonian) partial differential equations (PDEs), the resulting problem may be stiffly-oscillatory, rather than highly-oscillatory. In such a case, a different implementation of the methods is needed, in order to gain the maximum efficiency.Comment: 17 pages, 3 figure

Arbitrarily high-order energy-preserving methods for simulating the gyrocenter dynamics of charged particles

Gyrocenter dynamics of charged particles plays a fundamental role in plasma physics. In particular, accuracy and conservation of energy are important features for correctly performing long-time simulations. For this purpose, we here propose arbitrarily high-order energy conserving methods for its simulation. The analysis and the efficient implementation of the methods are fully described, and some numerical tests are reported.Comment: 23 pages, 4 figure

Spectral solution of ODE-IVPs by using SHBVMs

Recently, Hamiltonian Boundary Value Methods (HBVMs), have been used as spectral methods in time for effectively solving multi-frequency, highly-oscillatory and/or stiffly-oscillatory problems. A complete analysis of their use in such a fashion has been also carried out, providing a theoretical framework explaining their effectiveness. We report here a few numerical examples showing their potentialities to provide a fully accurate solver for general ODE problems