Asymptotics of a Time-Splitting Scheme for the Random Schrödinger Equation with Long-Range Correlations

Gomez, Christophe; Pinaud, Olivier

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 2, pp. 411-431.

This work is concerned with the asymptotic analysis of a time-splitting scheme for the Schrödinger equation with a random potential having weak amplitude, fast oscillations in time and space, and long-range correlations. Such a problem arises for instance in the simulation of waves propagating in random media in the paraxial approximation. The high-frequency limit of the Schrödinger equation leads to different regimes depending on the distance of propagation, the oscillation pattern of the initial condition, and the statistical properties of the random medium. We show that the splitting scheme captures these regimes in a statistical sense for a time stepsize independent of the frequency.

Classification : 65M12, 65M70, 65C30, 60H15
Mots clés : random Schrödinger equation, long-range correlations, high frequency asymptotics, splitting scheme

[1] G. Bal, T. Komorowski and L. Ryzhik, Kinetic limits for waves in a random medium. Kinet. Relat. Models 3 (2010) 529-644. | MR | Zbl

[2] G. Bal, T. Komorowski and L. Ryzhik, Asymptotics of the phase of the solutions of the random schrödinger equation. ARMA (2011) 13-64. | Zbl

[3] G. Bal and L. Ryzhik, Time splitting for wave equations in random media. ESAIM: M2AN 38 (2004) 961-988. | Numdam | MR | Zbl

[4] W. Bao, S. Jin and P.A. Markowich, On Time-Splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175 (2002) 487-524. | MR | Zbl

[5] P. Billingsley, Convergence of Probability Measures. John Wiley and Sons, New York (1999). | MR | Zbl

[6] S. Dolan, C. Bean and B. Riollet, The broad-band fractal nature of heterogeneity in the upper crust from petrophysical logs. Geophys. J. Int. 132 (1998) 489-507.

[7] J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave propagation and time reversal in randomly layered media, in vol. 56 of Stoch. Model. Appl. Probab. Springer, New York (2007). | MR

[8] C. Gomez, Radiative transport limit for the random Schrödinger equation with long-range correlations. J. Math. Pures. Appl. 98 (2012) 295-327. | MR | Zbl

[9] C. Gomez, Wave decoherence for the random Schrödinger equation with long-range correlations. To appear in CMP (2012). | MR | Zbl

[10] A.A. Gonoskov and I.A. Gonoskov, Suppression of reflection from the grid boundary in solving the time-dependent Schroedinger equation by split-step technique with fast Fourier transform, ArXiv Physics e-prints (2006).

[11] S. Jin, P. Markowich and C. Sparber, Mathematical and computational methods for semiclassical Schrödinger equations. Acta Numer. 20 (2011) 121-209. | MR | Zbl

[12] P.-L. Lions and T. Paul, Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9 (1993) 553-618. | MR | Zbl

[13] P.A. Markowich, P. Pietra and C. Pohl, Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit. Numer. Math. 81 (1999) 595-630. | MR | Zbl

[14] J.M. Martin and M. Flatté, Intensity images and statistics from numerical simulation of the wave propagation in 3-d random media. Appl. Optim. 247 (1988) 2111-2126.

[15] R.I. Mclachlan and G.R.W. Quispel, Splitting methods. Acta Numer. 11 (2002) 341-434. | MR | Zbl

[16] C. Sidi and F. Dalaudier, Turbulence in the stratified atmosphere: Recent theoretical developments and experimental results. Adv. Space Research 10 (1990) 25-36.

[17] G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506-517. | MR | Zbl

[18] F. Tappert, The parabolic approximation method, Wave propagation in underwater acoustics. In vol. 70 of Lect. Notes Phys. Springer (1977) 224-287. | MR