The aim of the project P-24157-N13 supported by the Austrian Science Fund (FWF)
is to investigate the numerical solution of nonlinear Schrödinger equations.
For the full discretization of initial value problems for these evolution equations we analyse
the method of lines approach, which is suitable for problems where theoretical insight suggests
a suitable space discretization and mesh. In this approach the evolutionary PDE is reduced to a
(generally large) system of initial value problems for ordinary differential equations.
For the resulting problems, splitting methods of high order will be investigated with respect to
the structure of the local error and implications for the adaptive choice of time steps. Furthermore,
new a priori and a posteriori error estimates based on the defect correction principle are put forward
for the approximation of the matrix exponential function by splitting methods. This task occurs as a
subproblem in the time integration of evolution equations in the context of the method of lines.
A realization of this approach for nonlinear evolution equations will represent a nontrivial extension.
Application of the project results to nonlinear single-particle Schrödinger equations associated with
model reductions of the high-dimensional linear multi-particle Schrödinger equation like time-dependent
density functional theory or the multi-configuration time-dependent Hartree-Fock method shall put the
project results in an application-oriented context. Collaborators in the project are
Winfried Auzinger (Vienna Universiyty of Technology),
Gustaf Söderlind (University of Lund), and
Mechthild Thalhammer (University of Innsbruck).
So far, the project yielded the following articles and preprints and
conference presentations. A movie illustrating the results of
our research so far can be found here.
W. Auzinger, O. Koch, M. Thalhammer,
Defect-based local error estimators for splitting methods, with application to Schrödinger equations,
Part I. The linear case,
J. Comput. Appl. Math. 236(2012), pp. 2643-2659.
H. Hofstätter, O. Koch, and M. Thalhammer,
Convergence of split-step generalized-Laguerre-Fourier-Hermite methods for Gross-Pitaevskii equations with rotation term,
Numer. Math. 127(2014), pp. 315-364.
W. Auzinger, O. Koch, M. Thalhammer,
Defect-based local error estimators for splitting methods, with application to Schrödinger equations,
Part II. Higher-order methods for linear problems,
J. Comput. Appl. Math. 255(2014), pp. 384-403.
W. Auzinger, H. Hofstätter, O. Koch, and M. Thalhammer,
Defect-based local error estimators for splitting methods, with application to Schrödinger equations,
Part III. The nonlinear case,
J. Comput. Appl. Math.
273(2014), pp. 182-204.
W. Auzinger,
O. Koch,
M. Thalhammer,
Defect-based local error estimators for high-order splitting methods involving
three linear operators,
Numer. Algorithms
70(2015), pp. 61-91.
W. Auzinger,
H. Hofstätter,
D. Ketcheson,
and O. Koch,
Practical splitting methods for the adaptive integration of nonlinear evolution equations.
Part I: Construction of optimized schemes and pairs of schemes,
BIT
57(2017), pp. 55-74,
DOI=dx.doi.org/10.1007/s10543-016-0626-9.
W. Auzinger,
Th. Kassebacher, O. Koch,
M. Thalhammer,
Adaptive splitting methods for nonlinear Schrödinger equations in the semiclassical regime,
Numer. Algorithms 72(2016), pp. 1-35.
W. Auzinger,
H. Hofstätter, and O. Koch,
Symbolic Manipulation of Flows of Nonlinear
Evolution Equations, with Application in the
Analysis of Split-Step Time Integrators, Springer LNCS 9890(2016), pp. 43-57.
O. Koch, joint with W. Auzinger, H. Hofstätter and M. Thalhammer,
Adaptive Full Discretization of Nonlinear Schrödinger Equations,
8th Austrian NA Day, Vienna, May 2012.
H. Hofstätter, joint with W. Auzinger, O. Koch and M. Thalhammer,
Convergence Analysis of a Generalized-Laguerre-Fourier-Hermite Method for the Gross-Pitaevskii Equation with Rotation Term,
8th Austrian NA Day, Vienna, May 2012.
O. Koch, joint with W. Auzinger, H. Hofstätter and M. Thalhammer,
Adaptive Full Discretization of Gross-Pitaevskii Equations with Rotation Term,
Workshop on Innovative Time Integration, Innsbruck, May 2012.
H. Hofstätter, joint with W. Auzinger, O. Koch and M. Thalhammer,
Convergence Analysis of a Generalized-Laguerre-Fourier-Hermite Method for the Gross-Pitaevskii Equation with Rotation Term,
Workshop on Innovative Time Integration, Innsbruck, May 2012.
H. Hofstätter, joint with W. Auzinger, O. Koch and M. Thalhammer,
A Perl Program for the Symbolic Manipulation of Flows of Differential Equations and Its Application to the
Analysis of Defect-Based Error Estimators for Splitting Methods,
Colloquium of the Department of Mathematics,
University of Innsbruck, January 2013.
H. Hofstätter, joint with W. Auzinger, O. Koch and M. Thalhammer,
A Perl Program for the Symbolic Manipulation of Flows of Differential Equations and Its Application to
the Analysis of Defect-Based Error Estimators for Splitting Methods,
9th Austrian NA Day, Graz, April 2013.
O. Koch, joint with W. Auzinger, H. Hofstätter, and M. Thalhammer,
Local Estimates of the Time-Stepping Error for High-Order Splitting Methods,
25th Biennial Numerical Analysis Conference, Glasgow, U.K., June 2013.
O. Koch, joint with W. Auzinger, H. Hofstätter, and
M. Thalhammer.
Analysis of High-Order Adaptive Full Discretization for Nonlinear Schrödinger Equations,
Functional Differential Equations, Gdansk, Poland, September 2013.
O. Koch, joint with W. Auzinger, H. Hofstätter, and
M. Thalhammer.
Fully Discrete Splitting Methods for Rotating Bose-Einstein Condensates,
18th ÖMG Congress and Annual DMV Meeting, Innsbruck, Austria, September 2013.
O. Koch, joint with W. Auzinger, H. Hofstätter, B. Muite, M. Quell, and
M. Thalhammer.
Parallel Adaptive Splitting Methods for Nonlinear Evolution Equations,
10th Austrian Numerical Analysis Day, Vienna, Austria, May 2014.
W. Auzinger, joint with H. Hofstätter, O. Koch, and M. Thalhammer.
Representation and Estimation of Local Errors for Splitting Methods Involving two or three Parts,
8th NAI Workshop on Numerical Analysis of Evolution Equations, Innsbruck, October 2014.
Th. Kassebacher, joint with W. Auzinger,
O. Koch, and M. Thalhammer.
Adaptive Time Splitting for Nonlinear Schrödinger Equations in the Semiclassical Regime,
8th NAI Workshop on Numerical Analysis of Evolution Equations, Innsbruck, October 2014.
W. Auzinger,
Representation and estimation of the local error of higher-order exponential splitting schemes involving two or three sub-operators,
Mathematical Seminar, University of Tartu, Estonia, September 2014.
Joint with O. Koch and M. Thalhammer.
The movie below shows a simulation of the dynamics of a rotating Bose-Einstein condensate
in two space dimensions. The underlying equation is the Gross-Pitaevskii equation with
rotation term,
where the parameters are chosen as Ω=0.5, β=100, γ=0.8 and the external potential is V(x,y)=0.4 y^{2}.
The underlying discretization is based on 100 Laguerre and 128 Fourier basis functions and time integration is performed
by the Strang splitting with step-size 0.02.