Convergence of split-step generalized-Laguerre-Fourier-Hermite methods for Gross-Pitaevskii equations with rotation term

A convergence analysis for time-splitting generalized-Laguerre– Fourier–Hermite pseudo-spectral methods applied to time-dependent Gross– Pitaevskii equations with rotation term is given. The space discretization combines the generalized-Laguerre–Fourier spectral method with respect to the (x, y)-variables and the Hermite spectral method with respect to the zdirection. For the time integration exponential operator splitting methods are studied. Under suitable regularity requirements on the problem data spectral accuracy of the spatial discretization and the nonstiff convergence order for the time integrator is retained. Essential ingredients are a general functional analytic framework of abstract nonlinear evolution equations and fractional power spaces defined by the principal linear part, Sobolev-type inequalities in curved rectangles, and results on the asymptotical distribution of the nodes and weights associated with Gauß–Laguerre quadrature. The theoretical convergence estimate is confirmed by a numerical example. We acknowledge financial support by the Austrian Science Fund (FWF) under the projects P21620-N13 and P24157-N13. Harald Hofstätter (Corresponding author) Institute for Analysis and Scientific Computing, Vienna University of Technology Wiedner Hauptstrasse 8–10, A–1040 Wien, Austria E-mail: [email protected] Othmar Koch Institute for Analysis and Scientific Computing, Vienna University of Technology Wiedner Hauptstrasse 8–10, A–1040 Wien, Austria E-mail: [email protected] Mechthild Thalhammer Institut für Mathematik, Leopold–Franzens Universität Innsbruck Technikerstraße 13/VII, A–6020 Innsbruck, Austria E-mail: [email protected]