Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian

Kormann, Katharina; Holmgren, Sverker; Karlsson, Hans

doi:10.1016/j.jocs.2011.02.003

Cited by 8 publications

(13 citation statements)

References 38 publications

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“…Thereafter, we will review a technique proposed in [37] to estimate the temporal error only. This will be needed for the adaptive algorithm discussed in the subsequent section.…”

Section: A Posteriori Error Estimatementioning

confidence: 99%

“…we therefore use the idea of [37] to apply a posteriori error estimation theory separately on the individual time steps: The estimated error from the previous time step is treated as a perturbation of the starting value and the perturbation due to numerical propagation is given by the local remainder in each step. For the Magnus-Lanczos propagator, the first dismissed term of the Magnus expansion is used to estimate the Magnus error and for the Lanczos error we use the remainder term as proposed in [32,51].…”

Section: Estimate For the Temporal Errormentioning

confidence: 99%

“…This means the local perturbations simply add up to the global error. This idea has been tested in [37] and will be employed in the adaptive algorithm proposed in Section 4.…”

Section: Estimate For the Temporal Errormentioning

confidence: 99%

“…However, they cannot show any global error bounds. Duality-based a posteriori error estimates have also been discussed for time stepping in ordinary differential equations (ODE) [14] and applied to the semidiscretized Schrödinger equation in [37] to achieve an adaptive time-stepping method. In this paper, we use the temporal adaptivity proposed in [37] and enhance it by introducing spatial adaptivity that is optimized for the computation of some given functional of the error.…”

Section: Introductionmentioning

confidence: 99%

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A Time-Space Adaptive Method for the Schrödinger Equation

Kormann

2016

Commun. Comput. Phys.

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Abstract. In this paper, we present a discretization of the time-dependent Schrödinger equation based on a Magnus-Lanczos time integrator and high-order Gauss-Lobatto finite elements in space. A truncated Galerkin orthogonality is used to obtain dualitybased a posteriori error estimates that address the temporal and the spatial error separately. Based on this theory, a space-time adaptive solver for the Schrödinger equation is devised. An efficient matrix-free implementation of the differential operator, suited for spectral elements, is used to enable computations for realistic configurations. We demonstrate the performance of the algorithm for the example of matter-field interaction.

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“…Thereafter, we will review a technique proposed in [37] to estimate the temporal error only. This will be needed for the adaptive algorithm discussed in the subsequent section.…”

Section: A Posteriori Error Estimatementioning

confidence: 99%

Section: Estimate For the Temporal Errormentioning

confidence: 99%

“…This means the local perturbations simply add up to the global error. This idea has been tested in [37] and will be employed in the adaptive algorithm proposed in Section 4.…”

Section: Estimate For the Temporal Errormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Time-Space Adaptive Method for the Schrödinger Equation

Kormann

2016

Commun. Comput. Phys.

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“…Previous work, however, is mainly concerned with the derivation of a priori error bounds, but does not treat the construction of a posteriori error estimators which were successfully applied for instance for exponential operator splitting methods [5,6]. A posteriori error estimation and adaptive step selection for Magnus-type integrators is to our knowledge only discussed in [22], where classical Magnus integrators are endowed with a global error estimator based on integration of an adjoint problem as suggested in [14]. Alternatively to the Magnus-type approaches, splitting methods could be used to eliminate the time-dependence by freezing the independent variable and propagating it separately, see [10].…”

Section: Introduction and Overviewmentioning

confidence: 99%

A posteriori error estimation for Magnus-type integrators

et al. 2019

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We study high-order Magnus-type exponential integrators for large systems of ordinary differential equations defined by a time-dependent skew-Hermitian matrix. We construct and analyze defect-based local error estimators as the basis for adaptive stepsize selection. The resulting procedures provide a posteriori information on the local error and hence enable the accurate, efficient, and reliable time integration of the model equations. The theoretical results are illustrated on two numerical examples .

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Stable Difference Methods for Block-Oriented Adaptive Grids

et al. 2014

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In this paper, we present a block-oriented scheme for adaptive mesh refinement based on summation-by-parts (SBP) finite difference methods and simultaneous-approximation-term (SAT) interface treatment. Since the order of accuracy at SBP-SAT grid interfaces is lower compared to that of the interior stencils, we strive at using the interior stencils across block-boundaries whenever possible. We devise a stable treatment of SBP-FD junction points, i.e. points where interfaces with different boundary treatment meet. This leads to stable discretizations for more flexible grid configurations within the SBP-SAT framework, with a reduced number of SBP-SAT interfaces. Both first and second derivatives are considered in the analysis. Even though the stencil order is locally reduced close to numerical interfaces and corner points, numerical simulations show that the locally reduced accuracy does not severely reduce the accuracy of the time propagated numerical solution. Moreover, we explain how to organize the grid and how to automatically adapt the mesh, aiming at problems of many variables. Examples of adaptive grids are demonstrated for the simulation of the time-dependent Schrödinger equation and for the advection equation.

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Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian

Cited by 8 publications

References 38 publications

A Time-Space Adaptive Method for the Schrödinger Equation

A Time-Space Adaptive Method for the Schrödinger Equation

A posteriori error estimation for Magnus-type integrators

Stable Difference Methods for Block-Oriented Adaptive Grids

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