Error analysis of splitting methods for the time dependent Schrödinger equation
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comunitat-uji-handle2:10234/7037
comunitat-uji-handle3:10234/8635
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Title
Error analysis of splitting methods for the time dependent Schrödinger equationDate
2011Publisher
Society for Industrial and Applied MathematicsISSN
1064-8275; 1095-7197Bibliographic citation
SIAM Journal on Scientific Computing (2011) vol. 33, no. 4, p. 1525-1548Type
info:eu-repo/semantics/articlePublisher version
http://epubs.siam.org/sisc/resource/1/sjoce3/v33/i4/p1525_s1Version
info:eu-repo/semantics/publishedVersionAbstract
A typical procedure to integrate numerically the time dependent Schrödinger equation involves two stages. In the first stage one carries out a space discretization of the continuous problem. This results in the linear ... [+]
A typical procedure to integrate numerically the time dependent Schrödinger equation involves two stages. In the first stage one carries out a space discretization of the continuous problem. This results in the linear system of differential equations idu/dt = Hu, where H is a real symmetric matrix, whose solution with initial value u(0) = u0 ∈ CN is given by u(t) = e−itHu0. Usually, this exponential matrix is expensive to evaluate, so that time stepping methods to construct approximations to u from time tn to tn+1 are considered in the second phase of the procedure. Among them, schemes involving multiplications of the matrix H with vectors, such as Lanczos and Chebyshev methods, are particularly efficient. In this work we consider a particular class of splitting methods which also involves only products Hu. We carry out an error analysis of these integrators and propose a strategy which allows us to construct different splitting symplectic methods of different order (even of order zero) possessing a large stability interval that can be adapted to different space regularity conditions and different accuracy ranges of the spatial discretization. The validity of the procedure and the performance of the resulting schemes are illustrated in several numerical examples. [-]
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