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.

Refereed Papers


• 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.
  
  You can download a preprint of this article as a .pdf file. Archived at https://uscholar.univie.ac.at/.
  
  
• H. Hofstätter, O. Koch, and M. Thalhammer, Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross–Pitaevskii equations, Numer. Math. 127(2014), pp. 315-364.
  
  You can download a preprint of this article as a .pdf file. Archived at https://uscholar.univie.ac.at/.
  
  
• 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.
  
  You can download a preprint of this article as a .pdf file. Archived at https://uscholar.univie.ac.at/.
  
  
• 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.
  
  You can download a preprint of this article as a .pdf file. Archived at https://uscholar.univie.ac.at/.
  
  
• 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.
  
  You can download a preprint of this paper as a .pdf file. Archived at https://uscholar.univie.ac.at/.
  
  
  
• 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.
  
  You can download a preprint of this paper as a .pdf file. Also available from http://arxiv.org/abs/1604.01179
  
  
  
• 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.
  
  You can download a preprint of this paper as a .pdf file. Also available from http://arxiv.org/abs/1605.00429. Archived at https://uscholar.univie.ac.at.
  
  
  
• 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.
  
  You can download a preprint of this paper from http://arxiv.org/abs/1605.00453
  
  
  
• W. Auzinger, W. Herfort, H. Hofstätter, and O. Koch, Setup of Order Conditions for Splitting Methods, Springer LNCS 9890(2016), pp. 30-42.
  
  You can download a preprint of this paper from http://arxiv.org/abs/1605.00445
  
  
  
• W. Auzinger, O. Koch, and M. Quell, Adaptive High-Order Splitting Methods for Systems of Nonlinear Evolution Equations with Periodic Boundary Conditions, Numer. Algorithms 75(2017), pp. 261-283. DOI=dx.doi.org/10.1007/s11075-016-0206-8.
  
  You can download a preprint of this paper as a .pdf file. Also available from http://arxiv.org/abs/1604.01201
  
  
  
• W. Auzinger, Th. Kassebacher, O. Koch, M. Thalhammer, Convergence of a Strang Splitting Finite Element Discretization for the Schrödinger-Poisson Equation, M2AN - Math. Model. Numer. Anal. 51(2017), pp. 1245-1278. DOI=dx.doi.org/10.1051/m2an/2016059.
  
  You can download a reprint of this paper as a .pdf file. The original publication is available at www.esaim-m2an.org, the copyright is owned by EDP Sciences. Preprint also available from http://arxiv.org/abs/1605.00437
  
  
  
• W. Auzinger, H. Hofstätter, O. Koch, and M. Thalhammer, Reduced Order of the Local Error Of Splitting For Parabolic Problems with Dirichlet Boundary Conditions, in `Numerical Analysis and Applied Mathematics' AIP Conference Proceedings1863, 160008 (2017). DOI=dx.doi.org/10.1063/1.4992342
  
  
  Oral Presentations
  
  
• 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.
  
  
• O. Koch, joint with W. Auzinger, H. Hofstätter and M. Thalhammer, Adaptive High-order Splitting Methods for Schrödinger Equations, SDIDE 2013 - Stability and Discretization Issues in Differential Equations, Pamplona, Spain, April 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.
  
  
• O. Koch, joint with W. Auzinger, H. Hofstätter, B. Muite, M. Quell, and M. Thalhammer. Parallel Adaptive Splitting Methods for Nonlinear Evolution Equations, International Workshop on Algorithms and Software for Scientific Computing, Vienna, Austria, July 2014.
  
  
• W. Auzinger, joint with O. Koch and M. Thalhammer. Representation and Estimation of the Local Error of Higher-Order Exponential Splitting Schemes Involving two or three Sub-Operators, 12th International Conference on Numerical Analysis and Applied Mathematics, Rhodes, September 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.
  
  
• Th. Kassebacher, Adaptive Time Splitting Methods for Nonlinear Schrödinger Equations in the Semiclassical Regime, 11th Austrian Numerical Analysis Day, Linz, May 2015. Joint with W. Auzinger, O. Koch, and M. Thalhammer.
  
  
• H. Hofstätter, On Splitting Methods for Nonlinear Parabolic Evolution Equations, 11th Austrian Numerical Analysis Day, Linz, May 2015. Joint with W. Auzinger, O. Koch, and M. Thalhammer.
  
  
• W. Auzinger, Splitting Schemes, Order Conditions, and Optimized Pairs, SciCADE - International Conference on Scientific Computing and Differential Equations 2015, Potsdam, Germany, September 2015. Joint with H. Hofstätter, D. Ketcheson, and O. Koch.
  
  
• T. Kassebacher, Convergence of Adaptive Splitting and Finite Element Methods for the Schrödinger-Poisson Equation, SciCADE - International Conference on Scientific Computing and Differential Equations 2015, Potsdam, Germany, September 2015. Joint with W. Auzinger, O. Koch, and M. Thalhammer.
  
  
• O. Koch, Adaptive High-order Time-Splitting Methods for Systems of Evolution Equations. Applications in Quantum Dynamics and Pattern Formation, SciCADE - International Conference on Scientific Computing and Differential Equations 2015, Potsdam, Germany, September 2015. Joint with W. Auzinger, H. Hofstätter, Th. Kassebacher, and M. Thalhammer.
  
  
• O. Koch, Adaptive Full Discretization for Nonlinear Evolution Equations, Innsbruck University, Austria, December 2015.
  
  
  Simulation Results
  
  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,

  i ∂t Ψ(x,y,t) = -½ Δ Ψ(x,y,t) +½ γ2(x2+y2)Ψ(x,y,t) +iΩ(x∂y-y∂x)Ψ(x,y,t) +V(x,y)Ψ(x,y,t)+β|Ψ(x,y,t)|2Ψ(x,y,t), (x,y)∈ ℝ2, t>0,	(1)
  Ψ(x,y,0)=Ψ0(x,y)=π-½(x+iy)e-½(x2+y2),	(2)
  Ψ ∈ H2,	(3)

  where the parameters are chosen as Ω=0.5, β=100, γ=0.8 and the external potential is V(x,y)=0.4 y². 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.
  
  Link to movie.

page written by Othmar Koch. last modification: Tue Oct 11 9:00 MET 2016