On the velocity space discretization for the Vlasov–Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods

Camporeale, Enrico; Delzanno, Gian Luca; Bergen, B.; Moulton, J. David

doi:10.1016/j.cpc.2015.09.002

Cited by 66 publications

(82 citation statements)

References 84 publications

(138 reference statements)

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“…[22] and employ a second order accurate Crank-Nicolson discretization scheme [33]. It will be shown in Sec.…”

Section: Temporal Discretizationmentioning

confidence: 99%

“…Indeed, in PIC it does not appear possible to conserve momentum and energy simultaneously for finite time step [2]. On the other hand recent work has shown that this is possible for spectral methods due to implicit time stepping [22]. In Ref.…”

Section: Introductionmentioning

confidence: 98%

“…For spectral methods, this is brought by the fact that one might need a large number of moments to capture the details of the distribution function. However, the continuous advance in computer power mitigates these limitations and continues to drive research in Eulerian Vlasov [12][13][14][15][16][17] and spectral methods [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

“…In Ref. [22] exact mass, momentum and energy conservation was achieved for the first time in one dimension and in the electrostatic limit (i.e. for the Vlasov-Poisson system) with finite time step (for reference, the numerical schemes discussed in Refs.…”

Section: Introductionmentioning

confidence: 99%

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Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

Delzanno

2015

Journal of Computational Physics

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A spectral method for the numerical solution of the multi-dimensional VlasovMaxwell equations is presented. The plasma distribution function is expanded in Fourier (for the spatial part) and Hermite (for the velocity part) basis functions, leading to a truncated system of ordinary differential equations for the expansion coefficients (moments) that is discretized with an implicit, second order accurate Crank-Nicolson time discretization. The discrete non-linear system is solved with a preconditioned Jacobian-Free Newton-Krylov method.It is shown analytically that the Fourier-Hermite method features exact conservation laws for total mass, momentum and energy in discrete form. Standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically. The whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities. Some preconditioning strategies are presented, showing that the number of linear iterations of the Krylov solver can be drastically reduced and a significant gain in performance can be obtained.

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“…[22] and employ a second order accurate Crank-Nicolson discretization scheme [33]. It will be shown in Sec.…”

Section: Temporal Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

Delzanno

2015

Journal of Computational Physics

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“…This approach has already been adopted in several works [14][15][16][17][18][19][20][21][22]. In the same spirit, here we focus on the one-dimensional LB operator.…”

Section: Introductionmentioning

confidence: 99%

Collisional effects on the numerical recurrence in Vlasov-Poisson simulations

2016

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The initial state recurrence in numerical simulations of the Vlasov-Poisson system is a well-known phenomenon. Here we study the effect on recurrence of artificial collisions modeled through the Lenard-Bernstein operator [A. Lenard and I. B. Bernstein, Phys. Rev. 112, 1456-1459(1958]. By decomposing the linear Vlasov-Poisson system in the Fourier-Hermite space, the recurrence problem is investigated in the linear regime of the damping of a Langmuir wave and of the onset of the bump-on-tail instability. The analysis is then confirmed and extended to the nonlinear regime through a Eulerian collisional Vlasov-Poisson code. It is found that, despite being routinely used, an artificial collisionality is not a viable way of preventing recurrence in numerical simulations without compromising the kinetic nature of the solution. Moreover, it is shown how numerical effects associated to the generation of fine velocity scales, can modify the physical features of the system evolution even in nonlinear regime. This means that filamentation-like phenomena, usually associated with low amplitude fluctuations contexts, can play a role even in nonlinear regime.

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On the Use of Hermite Functions for the Vlasov–Poisson System

Fatone

Funaro

Manzini

2020

Lecture Notes in Computational Science and Engineering

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We apply a second-order semi-Lagrangian spectral method for the Vlasov–Poisson system, by implementing Hermite functions in the approximation of the distribution function with respect to the velocity variable. Numerical tests are performed on a standard benchmark problem, namely the two-stream instability test case. The approach uses two independent sets of Hermite functions, based on Gaussian weights symmetrically placed with respect to the zero velocity level. An experimental analysis is conducted to detect a reasonable location of the two weights in order to improve the approximation properties.

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On the velocity space discretization for the Vlasov–Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods

Cited by 66 publications

References 84 publications

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

Collisional effects on the numerical recurrence in Vlasov-Poisson simulations

On the Use of Hermite Functions for the Vlasov–Poisson System

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