The development of an implicit, unconditionally stable, numerical method for solving the Vlasov–Poisson system in one dimension using a phase-space grid is presented. The algorithm uses the Crank–Nicolson discretization scheme and operator splitting allowing for direct solution of the finite difference equations. This method exactly conserves particle number, enstrophy and momentum. A variant of the algorithm which does not use splitting also exactly conserves energy but requires the use of iterative solvers. This algorithm has no dissipation and thus fine-scale variations can lead to oscillations and the production of negative values of the distribution function. We find that overall, the effects of negative values of the distribution function are relatively benign. We consider a variety of test cases that have been used extensively in the literature where numerical results can be compared with analytical solutions or growth rates. We examine higher-order differencing and construct higher-order temporal updates using standard composition methods.
Carrié, M. and Shadwick, B. A.. "An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system." Journal of Plasma Physics, vol. 88, no. 2, Mar. 2022. https://doi.org/10.1017/S0022377821001124
Carrié, M., & Shadwick, B. A. (2022). An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system. Journal of Plasma Physics, 88(2). https://doi.org/10.1017/S0022377821001124
Carrié, M., and Shadwick, B. A., "An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system," Journal of Plasma Physics 88, no. 2 (2022), https://doi.org/10.1017/S0022377821001124
@article{osti_1893545,
author = {Carrié, M. and Shadwick, B. A.},
title = {An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system},
annote = {The development of an implicit, unconditionally stable, numerical method for solving the Vlasov–Poisson system in one dimension using a phase-space grid is presented. The algorithm uses the Crank–Nicolson discretization scheme and operator splitting allowing for direct solution of the finite difference equations. This method exactly conserves particle number, enstrophy and momentum. A variant of the algorithm which does not use splitting also exactly conserves energy but requires the use of iterative solvers. This algorithm has no dissipation and thus fine-scale variations can lead to oscillations and the production of negative values of the distribution function. We find that overall, the effects of negative values of the distribution function are relatively benign. We consider a variety of test cases that have been used extensively in the literature where numerical results can be compared with analytical solutions or growth rates. We examine higher-order differencing and construct higher-order temporal updates using standard composition methods.},
doi = {10.1017/S0022377821001124},
url = {https://www.osti.gov/biblio/1893545},
journal = {Journal of Plasma Physics},
issn = {ISSN 0022-3778},
number = {2},
volume = {88},
place = {United Kingdom},
publisher = {Cambridge University Press (CUP)},
year = {2022},
month = {03}}
LASER-DRIVEN RELATIVISTIC PLASMAS APPLIED FOR SCIENCE, INDUSTRY, AND MEDICINE: The 1st International Symposium, AIP Conference Proceedingshttps://doi.org/10.1063/1.2958203