### EXPONENTIAL INTEGRATORS FOR STOCHASTIC SCHRÖDINGER EQUATIONS DRIVEN BY ITÔ NOISE

Rikard Anton, David Cohen

1. Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden Department of Mathematics, University of Innsbruck, A-6020 Innsbruck, Austria
• Received:2016-01-26 Revised:2016-11-08 Online:2018-03-15 Published:2018-03-15
• Supported by:

This work was partially supported by UMIT Research Lab at Umeå University and the Swedish Research Council (VR)(project nr.2013 -4562).

Rikard Anton, David Cohen. EXPONENTIAL INTEGRATORS FOR STOCHASTIC SCHRÖDINGER EQUATIONS DRIVEN BY ITÔ NOISE[J]. Journal of Computational Mathematics, 2018, 36(2): 276-309.

We study an explicit exponential scheme for the time discretisation of stochastic Schrödinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schrödinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.

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 [1] R. Anton, and D.Cohen, and S.Larsson, and X. Wang, Full discretisation of semi-linear stochastic wave Equations Driven by Multiplicative Noise, SIAM J. Numer. Anal., 54:2(2016), 1093-1119.[2] M. Barton-Smith, and A. Debussche, and L. Di Menza, Numerical study of two-dimensional stochastic NLS equations, Numer. Methods Partial Differential Equations, 21:4(2005), 810-842.[3] H. Berland, and A.L. Islas, and C.M. Schober, Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation, J. Comput. Phys., 225:1(2007), 284-299,[4] A. de Bouard, and A. Debussche, Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation, Appl. Math. Optim., 54:3(2006), 369-399.[5] A. de Bouard, and A. Debussche, The stochastic nonlinear Schrödinger equation in H 1, Stochastic Anal. Appl., 21:1(2003), 97-126.[6] A. de Bouard, and A. Debussche, On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation, Probab. Theory Related Fields, 123:1(2002), 76-96.[7] A. de Bouard, and A. Debussche, A stochastic nonlinear Schrödinger equation with multiplicative noise, Comm. Math. Phys., 205:1(1999), 161-181.[8] A. de Bouard, and A. Debussche, and L. Di Menza, Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations, Monte Carlo Methods Appl., 7:1-2(2001), 55-63.[9] T. Cazenave, and A. Haraux, An Introduction to Semilinear Evolution Equations, The Clarendon Press, Oxford University Press, New York. 1998[10] E. Celledoni, and D. Cohen, and B. Owren, Symmetric exponential integrators with an application to the cubic Schr dinger equation, Found. Comput. Math., 8:3(2008), 303-317.[11] D. Cohen, and L. Gauckler, Exponential integrators for nonlinear Schrdinger equations over long times, BIT, 52:4(2012), 877-903.[12] D. Cohen, and S. Larsson, and M. Sigg, A trigonometric method for the linear stochastic wave equation, SIAM J. Numer. Anal., 51:1(2013), 204-222,[13] D. Cohen, and L. Quer-Sardanyons, A fully discrete approximation of the one-dimensional stochastic wave equation, IMA J. Numer. Anal., 36:1(2016), 400-420.[14] G. Da Prato, and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.[15] A. de Bouard, and A. Debussche, A semi-discrete scheme for the stochastic nonlinear Schrödinger equation, Numer. Math., 96:4(2004), 733-770.[16] A. Debussche, and L. Di Menza, Numerical simulation of focusing stochastic nonlinear Schrödinger equations, Phys. D, 162:3-4(2002), 131-154.[17] E. Hebey, Nonlinear Analysis on Manifolds:Sobolev Spaces and Inequalities, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.[18] M. Hochbruck, and C. Lubich, Exponential integrators for quantum-classical molecular dynamics, BIT, 39:4(1999), 620-645.[19] M. Hochbruck, and A. Ostermann, Exponential integrators, Acta Numer., 19(2010), 209-286.[20] A. Jentzen, and P.E. Kloeden, The numerical approximation of stochastic partial differential equations, Milan J. Math., 77(2009), 205-244.[21] J. Liu, A mass-preserving splitting scheme for the stochastic Schrödinger equation with multiplicative noise, IMA J. Numer. Anal., 33:4(2013), 1469-1479[22] J. Liu, Order of convergence of splitting schemes for both deterministic and stochastic nonlinear Schrödinger equations, SIAM J. Numer. Anal., 51:4(2013), 1911-1932.[23] G.J. Lord, and J. Rougemont, A numerical scheme for stochastic PDEs with Gevrey regularity, IMA J. Numer. Anal., 24:4(2004), 587-604.[24] G.J. Lord, and A. Tambue, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise, IMA J. Numer. Anal., 33:2(2013), 515-543.[25] C. Prévôt, and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, Berlin, 2007.[26] X. Wang, An exponential integrator scheme for time discretization of nonlinear stochastic wave equation, J. Sci. Comput., (2014), 1-30.[27] B. Cano, and A. González-Pachón, Exponential time integration of solitary waves of cubic Schrödinger equation, Appl. Numer. Math., 91(2015), 26-45.[28] G. Dujardin, Exponential Runge-Kutta methods for the Schrödinger equation, Appl. Numer. Math., 59:8(2009), 1839-1857.[29] H. Berland, and B. Owren, and B. Skaflestad, Solving the nonlinear Schrdinger equation using exponential integrators, Modeling, Identification and Control, 27:4(2006), 201-218.[30] G. Dujardin, and E. Faou, Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential, Numer. Math., 108:2(2007), 223-262.[31] A. de Bouard, and A. Debussche, Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise, Ann. Probab., 33:3(2005), 1078-1110.[32] C.M. Mora, and R. Rebolledo, Regularity of solutions to linear stochastic Schrödinger equations, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 10:2(2007), 237-259.[33] S. Jiang, and L. Wang, and J. Hong, Stochastic multi-symplectic integrator for stochastic nonlinear Schrödinger equation, Commun. Comput. Phys., 14:2(2013), 393-411.[34] A.H. Strømmen Melbø, and D.J. Higham, Numerical simulation of a linear stochastic oscillator with additive noise, Appl. Numer. Math., 51:1(2004), 89-99.
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