A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System Based on Fourier, Legendre and Hermite Polynomials

doi:10.1007/s42967-019-00027-8

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (3): 333-360.doi: 10.1007/s42967-019-00027-8

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A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System Based on Fourier, Legendre and Hermite Polynomials

Lorella Fatone¹, Daniele Funaro², Gianmarco Manzini³

1 Dipartimento di Matematica, Università degli Studi di Camerino, Camerino, Italy;
2 Dipartimento di Scienze Chimiche e Geologiche, Università degli Studi di Modena e Reggio Emilia, Modena, Italy;
3 Group T-5, Applied Mathematics and Plasma Physics, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

收稿日期:2018-06-18 修回日期:2018-12-31 出版日期:2019-09-20 发布日期:2019-09-09
通讯作者: Daniele Funaro, Lorella Fatone, Gianmarco Manzini E-mail:daniele.funaro@unimore.it;lorella.fatone@unicam.it;gmanzini@lanl.gov
基金资助:
The second author was partially supported by the Short Term Mobility Program of the Consiglio Nazionale delle Ricerche (CNR-Italy). The third author was supported by the Laboratory Directed Research and Development Program (LDRD), U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences, and the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research, under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy by Los Alamos National Laboratory, operated by Los Alamos National Security LLC under contract DE-AC52-06NA25396.

A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System Based on Fourier, Legendre and Hermite Polynomials

Lorella Fatone¹, Daniele Funaro², Gianmarco Manzini³

1 Dipartimento di Matematica, Università degli Studi di Camerino, Camerino, Italy;
2 Dipartimento di Scienze Chimiche e Geologiche, Università degli Studi di Modena e Reggio Emilia, Modena, Italy;
3 Group T-5, Applied Mathematics and Plasma Physics, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

Received:2018-06-18 Revised:2018-12-31 Online:2019-09-20 Published:2019-09-09
Supported by:
The second author was partially supported by the Short Term Mobility Program of the Consiglio Nazionale delle Ricerche (CNR-Italy). The third author was supported by the Laboratory Directed Research and Development Program (LDRD), U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences, and the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research, under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy by Los Alamos National Laboratory, operated by Los Alamos National Security LLC under contract DE-AC52-06NA25396.

摘要/Abstract

摘要： In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF timestepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on some standard benchmark problems including the two-stream instability and the Landau damping test cases. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.

关键词: Spectral methods, Semi-Lagrangian methods, High-order, Hermite functions, Vlasov-Poisson equations, Mass conservation

Abstract: In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF timestepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on some standard benchmark problems including the two-stream instability and the Landau damping test cases. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.

Key words: Spectral methods, Semi-Lagrangian methods, High-order, Hermite functions, Vlasov-Poisson equations, Mass conservation

中图分类号:

Lorella Fatone, Daniele Funaro, Gianmarco Manzini. A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System Based on Fourier, Legendre and Hermite Polynomials[J]. Communications on Applied Mathematics and Computation, 2019, 1(3): 333-360.

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A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System Based on Fourier, Legendre and Hermite Polynomials

A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System Based on Fourier, Legendre and Hermite Polynomials

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