On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
Expokit: a software package for computing matrix exponentials
ACM Transactions on Mathematical Software (TOMS)
A fourth-order commutator-free exponential integrator for nonautonomous differential equations
SIAM Journal on Numerical Analysis
Applied Numerical Mathematics
A First Course in the Numerical Analysis of Differential Equations
A First Course in the Numerical Analysis of Differential Equations
Introduction to High Performance Computing for Scientists and Engineers
Introduction to High Performance Computing for Scientists and Engineers
Time-averaging and exponential integrators for non-homogeneous linear IVPs and BVPs
Applied Numerical Mathematics
Magnus integrators for solving linear-quadratic differential games
Journal of Computational and Applied Mathematics
New, Highly Accurate Propagator for the Linear and Nonlinear Schrödinger Equation
Journal of Scientific Computing
Hi-index | 31.45 |
We discuss the numerical solution of the Schrodinger equation with a time-dependent Hamilton operator using commutator-free time-propagators. These propagators are constructed as products of exponentials of simple weighted sums of the Hamilton operator. Owing to their exponential form they strictly preserve the unitarity of time-propagation. The absence of commutators or other computationally involved operations allows for straightforward implementation and application also to large scale and sparse matrix problems. We explain the derivation of commutator-free exponential time-propagators in the context of the Magnus expansion, and provide optimized propagators up to order eight. An extensive theoretical error analysis is presented together with practical efficiency tests for different problems. Issues of practical implementation, in particular the use of the Krylov technique for the calculation of exponentials, are discussed. We demonstrate for two advanced examples, the hydrogen atom in an electric field and pumped systems of multiple interacting two-level systems or spins that this approach enables fast and accurate computations.