Name
Abstract | ||
|---|---|---|
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 $i du/dt = H u$, where $H$ is a real symmetric matrix, whose solution with initial value $u(0) = u_0 \in \mathbb{C}^N$ is given by $u(t) = \mathrm{e}^{-i t H} u_0$. Usually, this exponential matrix is expensive to evaluate, so that time stepping methods to construct approximations to $u$ from time $t_n$ to $t_{n+1}$ 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. |
Year | DOI | Venue |
|---|---|---|
2011 | 10.1137/100794535 | SIAM J. Scientific Computing |
Keywords | Field | DocType |
real symmetric matrix,h u,different order,error analysis,order zero,different space regularity condition,different splitting symplectic method,splitting methods,time dependent schr,dinger equation,different accuracy range,typical procedure,exponential matrix,space discretization,symmetric matrix,differential equation,numerical analysis,time dependent schrodinger equation | Differential equation,Discretization,Mathematical optimization,Lanczos resampling,Matrix (mathematics),Mathematical analysis,Schrödinger equation,Symmetric matrix,Initial value problem,Matrix exponential,Mathematics | Journal |
Volume | Issue | ISSN |
33 | 4 | SIAM J. Sci. Comput. 33, No. 4 (2011), 1525-1548 |
Citations | PageRank | References |
3 | 0.78 | 3 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Sergio Blanes | 1 | 52 | 10.17 |
| Fernando Casas | 2 | 74 | 18.30 |
| A. Murua | 3 | 110 | 25.21 |