On the effectiveness of spectral methods for the numerical solution of multi-frequency highly oscillatory Hamiltonian problems

Multi-frequency, highly oscillatory Hamiltonian problems derive from the mathematical modelling of many real-life applications. We here propose a variant of Hamiltonian Boundary Value Methods (HBVMs), which is able to efficiently deal with the numerical solution of such problems. We present algorithms to select the parameters of the methods that allow one to obtain numerical approximations with spectral accuracy. We also propose an efficient implementation of the methods when using a Newton-type iteration to solve the implicit equations associated with this class of formulas.