In this paper, we consider a Lie splitting scheme for a nonlinear partial differential equation driven by a random time-dependent dispersion coefficient. Our main result is a uniform estimate of the error of the scheme when the time step goes to 0. Moreover, we prove that the scheme satisfies an asymptotic-preserving property. As an application, we study the order of convergence of the scheme when the dispersion coefficient approximates a (multi)fractional process.
Accepté le :
DOI : 10.1051/ps/2016023
Mots-clés : Nonlinear partial differential equations, splitting, stochastic partial differential equations, asymptotic-Preserving schemes, fractional and multifractional processes
@article{PS_2016__20__572_0,
author = {Duboscq, Romain and Marty, Renaud},
title = {Analysis of a splitting scheme for a class of random nonlinear partial differential equations},
journal = {ESAIM: Probability and Statistics},
pages = {572--589},
publisher = {EDP-Sciences},
volume = {20},
year = {2016},
doi = {10.1051/ps/2016023},
mrnumber = {3607207},
zbl = {1358.35168},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2016023/}
}
TY - JOUR AU - Duboscq, Romain AU - Marty, Renaud TI - Analysis of a splitting scheme for a class of random nonlinear partial differential equations JO - ESAIM: Probability and Statistics PY - 2016 SP - 572 EP - 589 VL - 20 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2016023/ DO - 10.1051/ps/2016023 LA - en ID - PS_2016__20__572_0 ER -
%0 Journal Article %A Duboscq, Romain %A Marty, Renaud %T Analysis of a splitting scheme for a class of random nonlinear partial differential equations %J ESAIM: Probability and Statistics %D 2016 %P 572-589 %V 20 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2016023/ %R 10.1051/ps/2016023 %G en %F PS_2016__20__572_0
Duboscq, Romain; Marty, Renaud. Analysis of a splitting scheme for a class of random nonlinear partial differential equations. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 572-589. doi : 10.1051/ps/2016023. http://www.numdam.org/articles/10.1051/ps/2016023/
G.P. Agrawal, Nonlinear Fiber Optics, 3rd edition. Academic Press, San Diego (2001). | Zbl
and , Time splitting for wave equations in random media. ESAIM: M2AN 38 (2004) 961–988 | DOI | Numdam | MR | Zbl
and , Time splitting for the Liouville equation in a random medium. Commun. Math. Sci. 2 (2004) 515–534. | DOI | MR | Zbl
, and , On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175 (2002) 487–524. | DOI | MR | Zbl
, and , Gaussian processes and Pseudodifferential Elliptic operators. Rev. Math. Iberoam. 13 (1997) 19–89. | MR | Zbl
, and , Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 26–40. | DOI | MR | Zbl
, and , An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit. Multiscale Model. Simul. 11 (2013) 1228–1260. | DOI | MR | Zbl
P. Billingsley, Convergence of Probability Measures. Wiley (1968). | MR | Zbl
and , Invariance principle, multifractional Gaussian processes and long-range dependence. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 475–489. | DOI | Numdam | MR | Zbl
and , Numerical passage from kinetic to fluid equations. SIAM J. Numer. Anal. 28 (1991) 26–42. | DOI | MR | Zbl
and , The nonlinear Schrodinger equation with white noise dispersion. J. Functional Anal. 259 (2010) 1300–1321. | DOI | MR | Zbl
and , 1D quintic nonlinear equation with white noise dispersion. J. Math. Pures Appl. 96 (2011) 363–376. | DOI | MR | Zbl
, Asymptotic-Preserving Schemes for Fluid Models of Plasmas. Panoramas et Synthèses 39-40 (2013) 1–90. | MR | Zbl
. Gaussian and their Subordinated Self-Similar Random Generalized Fields. Ann. Probab. 7 (1979) 1–28. | DOI | MR | Zbl
and , Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 (1979) 27–52. | DOI | MR | Zbl
P. Donnat, Quelques contributions mathématiques à l’optique non-linéaire. Ph.D. thesis, École Polytechnique (1993).
S.N. Ethier and T.G. Kurtz, Markov processes, characterization and convergence. Wiley, New York (1986). | MR | Zbl
J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Media. Springer (2007). | Zbl
, and , Relaxation schemes for nonlinear kinetic equations SIAM J. Numer. Anal. 34 (1997) 2168–2194. | DOI | MR | Zbl
and , Pulse propagation in random media with long range correlation. Multiscale Model. Simul. 7 (2009) 1302–1324. | DOI | MR | Zbl
and . Asymptotics of a time-splitting scheme for the random Schrödinger equation with long-range correlations. Math. Model. Numer. Anal. 48 (2014) 411–431. | DOI | Numdam | MR | Zbl
. Multiple Wiener integral. J. Math. Soc. Jpn 3 (1951) 157–169 | MR | Zbl
, Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations. SIAM J. Sci. Comput. 21 (1999) 441–454. | DOI | MR | Zbl
, and , Mathematical and computational methods for semiclassical Schrödinger equations. Acta Numer. 20 (2011) 121–209. | DOI | MR | Zbl
, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit. SIAM J. Numer. Anal. 35 (1998) 1073–1094. | DOI | MR | Zbl
, Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations. ESAIM: PS 9 (2005) 165–184. | DOI | Numdam | MR | Zbl
, On a splitting scheme for the nonlinear Schrödinger equation in a random medium. Commun. Math. Sci. 4 (2006) 679–705. | DOI | MR | Zbl
, From Hermite polynomials to multifractional processes. J. Appl. Prob. 50 (2013) 323–343. | DOI | MR | Zbl
and , A general framework for waves in random media with long-range correlations. Ann. Appl. Probab. 21 (2011) 115–139. | DOI | MR | Zbl
R.F. Peltier and J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results. preprint available at http://hal.inria.fr/inria-00074045/ (1995).
G. Samorodnitsky and M.S. Taqqu, Stable non-Gaussian random processes. Chapman and Hall (1994). | MR | Zbl
, On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506–517. | DOI | MR | Zbl
. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete. 31 (1975) 287–302. | DOI | MR | Zbl
. Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 (1979) 53–83. | DOI | MR | Zbl
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