Sixth-order schemes for laser–matter interaction in the Schrödinger equation

Singh, Pranav

doi:10.1063/1.5065902

Cited by 7 publications

(6 citation statements)

References 35 publications

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“…which is evaluated on the mesh and normalized to obtain a discrete starting vector v ∈ R n . This problem also appears in [29,51]. Similar to the previous subsection we discuss error estimates for the case p = 0, hence, the Krylov approximation of e −itB v. The implementation of the skew-Hermitian problem is described in Remark 4.…”

Section: Free Schrödinger Equation With a Double Well Potentialmentioning

confidence: 93%

A study of defect-based error estimates for the Krylov approximation of $φ$-functions

Jawecki

2020

Preprint

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Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential e tA v, is extended to the case of associated ϕ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g. for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the run. For other existing error estimates the reliability and performance is studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.

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Section: Free Schrödinger Equation With a Double Well Potentialmentioning

confidence: 93%

A study of defect-based error estimates for the Krylov approximation of $φ$-functions

Jawecki

2020

Preprint

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“…For the initial state of (5.2) we choose a Gaussian wave packet, ψ(t = 0, x) = (0.2π) −1/4 exp(−(x + 2.5) 2 /(0.4)), (5.3) which is evaluated on the mesh and normalized to obtain a discrete starting vector v ∈ R n . This problem also appears in [29,51]. We discuss error estimates for the case p = 0 (Krylov approximation of e −itB v).…”

Section: Free Schr öDinger Equation With a Double Well Potential And A Gaussian Wave Packet As An Initial Statementioning

confidence: 98%

A study of defect-based error estimates for the Krylov approximation of φ-functions

Jawecki

2021

Numer Algor

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Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential etAv, is extended to the case of associated φ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g., for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the fly. For other existing error estimates, the reliability and performance are studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.

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“…Remark 1. Methods based on a Strang splitting of the Magnus expansion were found to be very effective when the integrand is highly oscillatory [13,36]. Note that due to the special scalings of A and B, this splitting is actually of order four [13].…”

Section: Fourth Order Magnus-strang Splittingmentioning

confidence: 99%

Efficient Magnus-type integrators for solar energy conversion in Hubbard models

Auzinger,

Dubois,

Held

et al. 2021

Preprint

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Strongly interacting electrons in solids are generically described by Hubbardtype models, and the impact of solar light can be modeled by an additional time-dependence. This yields a finite dimensional system of ordinary differential equations (ODE)s of Schrödinger type, which can be solved numerically by exponential time integrators of Magnus type. The efficiency may be enhanced by combining these with operator splittings. We will discuss several different approaches of employing exponential-based methods in conjunction with an adaptive Lanczos method for the evaluation of matrix exponentials and compare their accuracy and efficiency. For each integrator, we use defect-based local error estimators to enable adaptive time-stepping. This serves to reliably control the approximation error and reduce the computational effort.

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Sixth-order schemes for laser–matter interaction in the Schrödinger equation

Cited by 7 publications

References 35 publications

A study of defect-based error estimates for the Krylov approximation of $φ$-functions

A study of defect-based error estimates for the Krylov approximation of $φ$-functions

A study of defect-based error estimates for the Krylov approximation of φ-functions

Efficient Magnus-type integrators for solar energy conversion in Hubbard models

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