Spectrally accurate space–time solution of Manakov systems

Abstract

In this paper, we study the numerical solution of Manakov systems by using a spectrally accurate Fourier decomposition in space, coupled with a spectrally accurate time integration. This latter relies on the use of spectral Hamiltonian boundary Value Methods. The used approach allows to conserve all the physical invariants of the systems. Some numerical tests are reported.

A general framework for solving differential equations

“…7 The coefficients of the polynomial approximation, in turn, are determined by imposing the orthogonality of the residual to a suitable polynomial space. In addition, we mention that the obtained methods can be also used as spectral methods in time, as was described in [20][21][22] in the case of Hamiltonian problems (see also [19,[44][45][46]). Future directions of investigation will concern the extension of this framework to different kinds of differential problems.…”

Continuous-Stage Runge-Kutta approximation to Differential Problems

“…A polynomial approximation of degree s (resulting, as usual, into an order 2s method) can be derived by formally substituting, in the equation (76), P ∞ (τ ) and I ∞ (τ ) with P s (τ ) and I s (τ ), respectively, 6 with the new approximations y (i) 1 ≈ y (i) (h), given by:…”

“…though different choices have been also considered [41]. 2 The arguments studied in this paper strictly follows those in [2] (in turn, inspired by [44,47,48]), derived from the energy-conserving methods called Hamiltonian Boundary Value Methods (HBVMs), which have been the subject of many investigations [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,19,20,17,21,22,23,26,24,25,27,28,29,30,32,31,33]. Here, we consider also a relevant generalization, w.r.t.…”

(Spectral) Chebyshev collocation methods for solving differential equations

“…HBVMs, in turn, can be also viewed as the outcome obtained after a local projection of the vector field onto a finite-dimensional function space: in particular, the set of polynomials of a given degree. For this purpose, the Legendre orthonormal polynomial basis has been considered so far [23] (see also [2]), and their use as spectral methods in time has been also investigated both theoretically and numerically [28,17,13,5,7,8]. Remarkably, as was already observed in [23], this idea is even more general and can be adapted to other finite-dimensional function spaces and/or different bases.…”

“…Here, we have used the name of the Matlab © functions implementing the two transformations 8. As is usual, 1 flop denotes a basic floating-point operation.…”