Appl Math Optim 54:369–399 (2006)
2006 Springer Science+Business Media, Inc.
Weak and Strong Order of Convergence of a Semidiscrete Scheme
for the Stochastic Nonlinear Schr¨odinger Equation
Anne de Bouard
and Arnaud Debussche
CNRS, Laboratoire de Math´ematiques, Universit´e de Paris-Sud,
91405 Orsay Cedex, France
IRMAR et ENS de Cachan, Antenne de Bretagne,
Campus de Ker Lann, Av. R. Schuman, 35170 Bruz, France
Abstract. In this article we analyze the error of a semidiscrete scheme for the
stochastic nonlinear Schr¨odinger equation with power nonlinearity. We consider
supercritical or subcritical nonlinearity and the equation can be either focusing
or defocusing. Allowing sufﬁcient spatial regularity we prove that the numerical
scheme has strong order
in general and order 1 if the noise is additive. Furthermore,
we also prove that the weak order is always 1.
Key Words. Nonlinear Schr¨odinger equations, Stochastic partial differential
equations, Numerical schemes, Rate of convergence.
AMS Classiﬁcation. 60H15, 35Q55, 60M15.
The numerical analysis of stochastic partial differential equations is a recent subject.
There are now a certain number of articles devoted to this ﬁeld but many problems still
need to be solved. The numerical analysis of schemes for stochastic differential equations
is much better understood. It started with the pioneering work of Milstein ,  and
Talay , . It is known that the Euler scheme is in general of strong order
order 1. Also in the case of an additive noise, it is of strong order 1. Various developments
can be found in the literature including the derivation of higher-order schemes (in the
weak or strong sense), and the expansion of the error in terms of the time step. (See for
instance , ,  and .)