Higher order operator splitting methods via Zassenhaus product formula: Theory and applications

Geiser, Jürgen; Tanoğlu, Gamze; Gücüyenen, Nurcan

doi:10.1016/j.camwa.2011.06.043

Cited by 18 publications

(11 citation statements)

References 13 publications

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“…The numerical method used in this work is the same numerical scheme described in [7,8], is based on the so called splitting method [20] to solve the proposed mathematical model defined by:…”

Section: Methodsmentioning

confidence: 99%

Discrete kinetic theory for 2D modeling of a moving crowd: Application to the evacuation of a non-connected bounded domain

Elmoussaoui

Argoul

Rhabi

et al. 2018

Computers & Mathematics with Applications

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This paper concerns the mathematical modeling of the motion of a crowd in a non connected bounded domain, based on the kinetic and stochastic game theories. The proposed model is derived by a hybrid approach at an intermediate mesoscopic representation between micro and macro-features; pedestrians interactions with various obstacles being managed from a probabilistic perspective. A proof of the existence and uniqueness of the proposed mathematical models solution is given for large times. A numerical resolution scheme based on the Splitting method is implemented and then applied to the crowd evacuation in a non connected bounded domain with one rectangular obstacle. The evacuation time of the room is then calculated by our technique, according to the dimensions and position of a square-shaped obstacle, and finally compared to that obtained by a deterministic approach by randomly varying some of its parameters.

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Section: Methodsmentioning

confidence: 99%

Discrete kinetic theory for 2D modeling of a moving crowd: Application to the evacuation of a non-connected bounded domain

Elmoussaoui

Argoul

Rhabi

et al. 2018

Computers & Mathematics with Applications

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“…A delicate problem in splitting methods is to achieve sufficient accuracy in the first splitting steps; see [10]. Based on the accuracy of initial starting solution, the results of all the next methods, for example MPE schemes or iterative scheme, are dependent and influenced by the order of the initialization process; see [11].…”

Section: Improving the Initialisation Of Splitting Methodsmentioning

confidence: 99%

“…Remark 9. The same idea can also be carried out using the Strang splitting method; see the linear case in [10]. We achieve a novel order of the new scheme to O( 1 + 2 ), 1 , 2 > 1, with 1 = 2 and 2 being the number of Zassenhaus exponents.…”

Section: Improving the Initialisation Of Splitting Methodsmentioning

confidence: 99%

Nonlinear Extension of Multiproduct Expansion Schemes and Applications to Rigid Bodies

Geiser

2013

International Journal of Differential Equations

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In this paper we discuss time integrators for nonlinear differential equations. In recent years, splitting approaches have become an important tool for reducing the computational time needed to solve differential equations. Moreover, nonlinearity is a challenge to splitting schemes, while one has to extend the exp-functions in terms of a nonlinear Magnus expansion. Here we discuss a novel extension of the so-called multiproduct expansion methods, which is used to improve the standard Strang splitting schemes as to their nonlinearity. We present an extension of linear splitting schemes and concentrate on nonlinear systems of differential equations and generalise in this respect the recent MPE method; see (Chin and Geiser, 2011). Some first numerical examples, of rigid body problems, are given as benchmarks.

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“…This can lead to non-phenomenological behavior of the system. Additional operator splitting techniques have been developed and include higher order methods such as Yoshida splitting (4 th and 6 th order), Kahan splitting, and Zassenhaus products 21,34,45,93 . Although the accuracy of the splitting method increases with these methods, additional function evaluations (some requiring steps backwards in time) make them more complicated approaches.…”

Section: Central Concepts For Hybrid Multi-scale Abmsmentioning

confidence: 99%

Strategies for Efficient Numerical Implementation of Hybrid Multi-scale Agent-Based Models to Describe Biological Systems

2014

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Biologically related processes operate across multiple spatiotemporal scales. For computational modeling methodologies to mimic this biological complexity, individual scale models must be linked in ways that allow for dynamic exchange of information across scales. A powerful methodology is to combine a discrete modeling approach, agent-based models (ABMs), with continuum models to form hybrid models. Hybrid multi-scale ABMs have been used to simulate emergent responses of biological systems. Here, we review two aspects of hybrid multi-scale ABMs: linking individual scale models and efficiently solving the resulting model. We discuss the computational choices associated with aspects of linking individual scale models while simultaneously maintaining model tractability. We demonstrate implementations of existing numerical methods in the context of hybrid multi-scale ABMs. Using an example model describing Mycobacterium tuberculosis infection, we show relative computational speeds of various combinations of numerical methods. Efficient linking and solution of hybrid multi-scale ABMs is key to model portability, modularity, and their use in understanding biological phenomena at a systems level.

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Higher order operator splitting methods via Zassenhaus product formula: Theory and applications

Cited by 18 publications

References 13 publications

Discrete kinetic theory for 2D modeling of a moving crowd: Application to the evacuation of a non-connected bounded domain

Discrete kinetic theory for 2D modeling of a moving crowd: Application to the evacuation of a non-connected bounded domain

Nonlinear Extension of Multiproduct Expansion Schemes and Applications to Rigid Bodies

Strategies for Efficient Numerical Implementation of Hybrid Multi-scale Agent-Based Models to Describe Biological Systems

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