A semi-discrete scheme for the stochastic nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: A semi-discrete scheme for the stochastic nonlinear Schrödinger equation
zbMATH Open document ID: 1055.65008
author: A. Debussche
author: A. de Bouard
published in: Numerische Mathematik
publication date: 2004-07-01T00:00:00Z
review text: The authors consider the multipicative nonlinear Schrödinger equation in the sense of Stratonovich \[ idu+(\Delta u+ \lambda| u|^{2\sigma} u)\,dt= u\circ dW,\quad u(0,x)= u_0(x), \] where \(x\in \mathbb{R}^d\), \(t\geq 0\), \(u(t,x)\in C\), \(\sigma> 0\), \(\lambda=\pm 1\), \(W\) is a real-valued Wiener process depending on \(t\), \(x\). For a fixed positive \(T_0\), they set \(\delta t= T_0/n\) for each natural \(n\) and introduce the scheme defined by the following semi-discrete equation \[ i{u_{l+1}- u_l\over\delta t}+\Delta u_{l+1/2}+ f(| u_l|,| u_{l+1}|) u_{l+1/2}= {\chi_l\over\sqrt{\delta t}} u_{l+1/2},\quad l= 0,\dots, n-1. \] Here \(u_l= u^n_l\), \(l= 0,\dots, n\), is an approximation of \(u(l\delta t)\), \[ u_{l+1/2}:= {1\over 2} (u_l+ u_{l+1}),\quad \chi_l:= {W((l+ 1)\delta t)- W(l\delta t)\over \sqrt{\delta t}},\quad l+ 0,\dots, n-1, \] and \[ f(a,b):= {\lambda\over \sigma+1} {a^{2\sigma+ 2}- b^{2\sigma+ 2}\over a^2- b^2}\text{ if }a\neq b,\;f(a,a)=\lambda a^{2\sigma}. \] For the scheme, the authors prove a convergence theorem.
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 60H35
zbMATH DE Number: 2077947
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: convergence
zbMATH Keywords: semidiscretization
zbMATH Keywords: nonlinear Schrödinger equation
reviewed by: Grigori N. Milstein
MaRDI profile type: Publication
full work available at URL: s00211-003-0494-5
OpenAlex ID: W1980882165
DOI: 10.1007/S00211-003-0494-5
description: scientific article; zbMATH DE number 2077947