Three-Dimensional Relativistic Particle-in-Cell Hybrid Code Based on an Exponential Integrator

Tückmantel, T.; Pukhov, A.; Liljo, Jalo; Hochbruck, Marlis

doi:10.1109/tps.2010.2056706

Cited by 15 publications

(25 citation statements)

References 25 publications

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“…It does not obey condition (11a) but only the weaker estimate |(cos(z) + 1)ψ E ( 1 2 z)| ≤ C 0 sinc( 1 2 z). Detailed numerical tests in Section 8.1 for this choice reveal sharp resonances a even multiples of π/ω, which where not observed in [15].…”

Section: Main Theoremmentioning

confidence: 82%

“…These rather restrictive assumptions make the model (1) linear. A more detailed derivation of the model may be found in [15,14]. Equation (1) has to be supplemented with boundary conditions and initial values.…”

Section: Physical Problem and Spatial Discretizationmentioning

confidence: 99%

“…[4,11,7,6,8]. We follow [13,15], introduce filter functions symmetrically and obtain the following numerical scheme…”

Section: Numerical Scheme and Filter Functionsmentioning

confidence: 99%

“…And hence it would be the plasma frequency that would impose a step size restriction in explicit Runge-Kutta or multistep methods. To overcome the restriction on the time step size due the plasma frequency a triple splitting method with filter functions was introduced by Liljo and Tückmantel, Pukhov, Liljo and Hochbruck in [13,15] for this model problem. An astute choice of filter functions results in a method that shows excellent behavior in numerical experiments.…”

Section: Introductionmentioning

confidence: 99%

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Convergence analysis of an explicit splitting method for laser plasma interaction simulations

Jansing¹,

Schädle²

2018

etna

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Abstract. The convergence of a triple splitting method originally proposed by Tückmantel, Pukhov, Liljo, and Hochbruck for the solution of a simple model that describes laser plasma interactions with overdense plasmas is analyzed. For classical explicit integrators it is the large density parameter that imposes a restriction on the time step size to make the integration stable. The triple splitting method contains an exponential integrator in its central component and was specifically designed for systems that describe laser plasma interactions and overcomes this restriction. We rigorously analyze a slightly generalized version of the original method. This analysis enables us to identify modifications of the original scheme such that a second-order convergent scheme is obtained.Key words. exponential integrators, highly oscillatory problems, trigonometric integrators, splitting methods AMS subject classifications. 65P101. Introduction. We consider the numerical solution of a system of equations describing laser plasma interactions with an overdense plasma that is a simplified version of the models considered in [14,16]. Essentially as in [17], the laser is described by Maxwell's equations, and the plasma is modeled as a fluid only, in contrast to [14,16]. After discretizing in space with a fixed spatial grid size h, a system of ordinary differential equations is obtained. This highly oscillatory system of ordinary differential equations is discretized in time by a triple splitting method with filter functions. The introduction of filter functions is a widely-used method to avoid resonance effects in splitting methods applied to oscillatory differential equations; see, e.g., [4, 6, 7, 11] and [8].The situation to consider now is slightly unusual as it is the localized overdense plasma and not the spatial discretization that gives rise to fast oscillations in the solution. The plasma frequency is several orders of magnitude larger than the laser frequency, which has to be resolved by the spatial grid. Hence, it is the plasma density ρ that imposes a step size restriction in explicit Runge-Kutta or multistep methods. To overcome the restriction on the time step size due to the plasma density, a triple splitting method with filter functions was introduced in [13,17] for this model problem. An astute choice of filter functions results in a method that shows excellent behavior in numerical experiments. The numerical experiments in [17] indicate convergence of second order in the time step size τ independent of ρ. A more detailed experiment, which is reported in Section 8.1, reveals that the method from [17] is not of second order in τ independent of the plasma density but is merely stable.By our convergence analysis of the triple splitting we are able to formulate conditions on the filter functions to obtain second-order convergence in τ independent of the plasma density ρ. These conditions can be fulfilled by slightly modifying the choice of the filter functions originally proposed in [13,17]. The modification comes ...

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Section: Main Theoremmentioning

confidence: 82%

Section: Physical Problem and Spatial Discretizationmentioning

confidence: 99%

“…[4,11,7,6,8]. We follow [13,15], introduce filter functions symmetrically and obtain the following numerical scheme…”

Section: Numerical Scheme and Filter Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Convergence analysis of an explicit splitting method for laser plasma interaction simulations

Jansing¹,

Schädle²

2018

etna

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“…To simulate the shot noise properly, the following method of initial particle positioning was developed and introduced into the hybrid 3D particle-in-cell code VLPL [28,29]. Since the number of numerical macroparticles is several orders of magnitude smaller than the 1:15 Â 10 11 protons in the SPS bunch, the noise generated by randomly seeded macroparticles would be significantly higher than the natural expectation (8).…”

mentioning

confidence: 99%

Natural noise and external wakefield seeding in a proton-driven plasma accelerator

Lotov

Lotova

Lotov

et al. 2013

Phys. Rev. ST Accel. Beams

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An accurate description of noise levels is of crucial importance for the correct simulation of instabilitydriven processes, such as the density modulation of a long proton bunch traversing a plasma. To insure that the correct instability develops, a seed field must be larger than the cumulative shot noise. We develop an analytical theory of the noise field and compare it with multidimensional simulations. We find that the natural noise wakefield generated in a plasma by the CERN Super Proton Synchrotron bunches is very low, at the level of 10 kV=m. This fortunate fact eases the requirements on the seed. Our threedimensional simulations show that even a few tens MeV electron bunch precursor of a very moderate intensity is sufficient to seed the proton bunch self-modulation in plasma.

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A particle-in-cell method for the simulation of plasmas based on an unconditionally stable field solver

Wolf

Causley

Christlieb

et al. 2016

Journal of Computational Physics

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We propose a new particle-in-cell (PIC) method for the simulation of plasmas based on a recently developed, unconditionally stable solver for the wave equation. This method is not subject to a CFL restriction, limiting the ratio of the time step size to the spatial step size, typical of explicit methods, while maintaining computational cost and code complexity comparable to such explicit schemes. We describe the implementation in one and two dimensions for both electrostatic and electromagnetic cases, and present the results of several standard test problems, showing good agreement with theory with time step sizes much larger than allowed by typical CFL restrictions.

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Three-Dimensional Relativistic Particle-in-Cell Hybrid Code Based on an Exponential Integrator

Cited by 15 publications

References 25 publications

Convergence analysis of an explicit splitting method for laser plasma interaction simulations

Convergence analysis of an explicit splitting method for laser plasma interaction simulations

Natural noise and external wakefield seeding in a proton-driven plasma accelerator

A particle-in-cell method for the simulation of plasmas based on an unconditionally stable field solver

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