Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: Construction of optimized schemes and pairs of schemes

Auzinger, Winfried; Hofstätter, Harald; Ketcheson, David I.; Koch, Othmar

doi:10.1007/s10543-016-0626-9

Cited by 29 publications

(63 citation statements)

References 22 publications

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Order By: Relevance

“…Ref. [60] and are based on reusing a number of evaluations from the basic integrator. In our simulations, we have used an embedded pair of orders 4(5) referred to as Emb 5/4 AK (ii) in the collection [61].…”

Section: Discussionmentioning

confidence: 99%

Long-time expansion of a Bose-Einstein condensate: Observability of Anderson localization

Donsa¹,

Hofstätter²,

Koch³

et al. 2017

Phys. Rev. A

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We numerically explore the long-time expansion of a one-dimensional Bose-Einstein condensate in a disorder potential employing the Gross-Pitaevskii equation. The goal is to search for unique signatures of Anderson localization in the presence of particle-particle interactions. Using typical experimental parameters we show that the time scale for which the non-equilibrium dynamics of the interacting system begins to diverge from the non-interacting system exceeds the observation times up to now accessible in the experiment. We find evidence that the long-time evolution of the wavepacket is characterized by (sub)diffusive spreading and a growing effective localization length suggesting that interactions destroy Anderson localization.

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Section: Discussionmentioning

confidence: 99%

Long-time expansion of a Bose-Einstein condensate: Observability of Anderson localization

Donsa¹,

Hofstätter²,

Koch³

et al. 2017

Phys. Rev. A

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“…giving the single additional condition 2 a 2 b 1 = 1 for order p ≥ 2. Assuming now that a 1 , a 2 and b 1 , b 2…”

Section: Example 2 ( [2]mentioning

confidence: 99%

“…By induction we conclude that the whole procedure is correct. See [2] for a more detailed exposition of this argument.…”

Section: The Algorithm: Implicit Recursive Eliminationmentioning

confidence: 99%

“…Topic (ii) is not discussed in this paper. Details concerning the theoretical background and a discussion concerning concrete results and optimized schemes obtained are given in [2], and a collection of optimized schemes can be found at [3]. We note that a related approach has recently also been considered in [5].…”

Section: Introductionmentioning

confidence: 99%

“…

This article is based on [1] and [2], where an approach based on Taylor expansion and the structure of its leading term as an element of a free Lie algebra was described for the setup of a system of order conditions for operator splitting methods. Along with a brief review of these materials and some theoretical background, we discuss the implementation of the ideas from these papers in computer algebra, in particular using 3 Maple 18.

…”

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confidence: 99%

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Setup of Order Conditions for Splitting Methods

Auzinger

Herfort

Hofstätter

et al. 2016

Computer Algebra in Scientific Computing

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A simple and efficient adaptive time stepping technique for low‐order operator splitting schemes applied to cardiac electrophysiology

Ogiermann

Perotti

Balzani

2023

Numer Methods Biomed Eng

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We present a simple, yet efficient adaptive time stepping scheme for cardiac electrophysiology (EP) simulations based on standard operator splitting techniques. The general idea is to exploit the relation between the splitting error and the reaction's magnitude—found in a previous one‐dimensional analytical study by Spiteri and Ziaratgahi—to construct the new time step controller for three‐dimensional problems. Accordingly, we propose to control the time step length of the operator splitting scheme as a function of the reaction magnitude, in addition to the common approach of adapting the reaction time step. This conforms with observations in numerical experiments supporting the need for a significantly smaller time step length during depolarization than during repolarization. The proposed scheme is compared with classical proportional–integral–differential controllers using state‐of‐the‐art error estimators, which are also presented in details as they have not been previously applied in the context of cardiac EP with operator splitting techniques. Benchmarks show that choosing the time step as a sigmoidal function of the reaction magnitude is highly efficient and full cardiac cycles can be computed with precision even in a realistic biventricular setup. The proposed scheme outperforms common adaptive time stepping techniques, while depending on fewer tuning parameters.

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Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: Construction of optimized schemes and pairs of schemes

Cited by 29 publications

References 22 publications

Long-time expansion of a Bose-Einstein condensate: Observability of Anderson localization

Long-time expansion of a Bose-Einstein condensate: Observability of Anderson localization

Setup of Order Conditions for Splitting Methods

A simple and efficient adaptive time stepping technique for low‐order operator splitting schemes applied to cardiac electrophysiology

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