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

Abstract

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.

Line Integral Solution of Differential Problems

“…The plot of such a function is depicted in Figure 2 for the double precision IEEE: as one may see, the function is well approximated by the line [27] 24 + 0.7 · ωh.…”

“…On the other hand, the application of the simplified Newton iteration for solving (34) reads, by considering that (see (27) and (28))…”

“…A common feature of the methods proposed so far, however, is that of requiring the use of time-steps h such that hω < 1, either for stability and/or accuracy requirements. We here sketch the approach recently defined in [27], relying on the use of HBVMs, which will allow the use of stepsizes h without such a restriction.…”

“…Assuming the ansatz f (y(ch)) ∼ e νJ 2 ⊗A ch , for a suitable ν ≥ 1 (essentially, one requires that ∇ f is well approximated by a polynomial of degree ν), the requirement (109) is accomplished by choosing ( [27], Criterion 2)…”

“…Further generalizations, along several directions, have been considered in [19][20][21][22][23][24][25][26][27][28]: in particular, in [19] Hamiltonian boundary value problems have been considered, which are not covered in this review. The main reference on line integral methods is given by the monograph [1].…”