Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross–Pitaevskii equations

Abstract

A convergence analysis of time-splitting pseudo-spectral methods adapted for time-dependent Gross-Pitaevskii equations with additional rotation term is given. For the time integration high-order exponential operator splitting methods are studied, and the space discretization relies on the generalized-Laguerre-Fourier spectral method with respect to the (x, y)variables as well as the Hermite spectral method in the z-direction. Essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations, fractional power spaces defined by the principal linear part, a Sobolev-type inequality in a curved rectangle, and results on the asymptotical distribution of the nodes and weights associated with Gauß-Laguerre quadrature. The obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained, provided that the problem data satisfy suitable regularity requirements. A numerical example confirms the theoretical convergence estimate.We acknowledge financial support by the Austrian Science Fund (FWF) under the projects P21620-N13 and P24157-N13. The final publication is available at Springer via

Convergence of multi-revolution composition time-splitting methods for highly oscillatory differential equations of Schrödinger type

“…The considerations can be extended to Schrödinger equations defined by a self-adjoint operator, employing the associated countable complete orthonormal system of eigenfunctions, see for instance [13,16,20]. The restriction to the linear case significantly reduces the complexity in the derivation of stability results and error expansions.…”

“…Making use of the fact that the considered differential operator A : D(A) → X is self-adjoint and positive semi-definite with pure point spectrum, permits to incorporate relevant problems of the form (2.3a) that are related to other spectral methods such as the Hermite or generalised Laguerre-Fourier-Hermite spectral method, see [13,16] and references therein. In this situation, standard results [20] ensure that the family of eigenfunctions forms a countable complete orthonormal system in the underlying Hilbert space.…”

Efficient time integration methods for Gross-Pitaevskii equations with rotation term

“…Given the relevance of the theme, numerous contributions are concerned with the development of advanced methodologies and illustrate their benefits over standard numerical methods; we in particular mention an approach based on generalised-Laguerre-Fourier-Hermite spectral space discretisation and operator splitting methods that has been studied in [9,22] and the transformation to the rotating frame combined with the application of exponential integrators as proposed in [5,6,7,10,11]. We point out that a rigorous stability and local error analysis has shown the reliability of generalised-Laguerre-Fourier-Hermite pseudo-spectral time-splitting methods, see [22]; but, due to the fact that the implementation of these specialised spectral methods is involved and computationally expensive, we believe that it is expedient to develop [10] further. In order to design higher-order schemes with improved accuracy and efficiency, we transform the considered time-dependent Gross-Pitaevskii equation to rotating Lagrangian coordinates; this yields a non-autonomous nonlinear Schrödinger equation of the form (6), where the afore sketched strategy can be realised by fast Fourier transforms.…”

Strang splitting for a semilinear Schrödinger equation with damping and forcing

“…The splitting approach can also be applied to the Gross-Pitaevskii equation, a NLS on R d with constraining polynomial potential, by using the basis of Hermite functions for the space discretization. The accuracy of such integrators has been analyzed, e.g., by [BBD02], [DT13], [Fao12], [Gau11], [HKT14], [KNT13], [Lub08], [ML15], and [Tha12]. The long-time behavior of numerical solutions, in particular the (near-)conservation of invariants over long times and the stability of plane waves, has been investigated by [Fao12], [FGL14], [GL10], and for exponential integrators by [CG12].…”

“…[DT13], [Fao12], [Gau11], [HKT14], [KNT13], [ML15], and [Tha12]. In contrast to these works, we avoid the notationally rather involved Lie derivatives, because in case of the LugiatoLefever equation the iterated commutators between the linear and nonlinear part are not the only source of error: An additional difficulty arising in our situation is the fact that the forcing term is coupled to the space derivatives and to the nonlinear part in a complicated way.…”