One-dimensional Vlasov-Poisson equation and its Hamiltonian fluid reductions

• Vlasov1D_4field.mlx: MATLAB live script to compute and represent the series expansion of the explicit closure S₄=S₄(S₂,S₃) and S₅=S₅(S₂,S₃); also compute the series expansions of the three Casimir invariants C₁, C₂ and C₃

• ParametricClosure4.nb: Mathematica notebook to check and represent the parametric closure S₂=S₂(Γ₂,Γ₃), S₃=S₃(Γ₂,Γ₃), S₄=S₄(Γ₂,Γ₃) and S₅=S₅(Γ₂,Γ₃)

• VP1D4f python code

  • VP1D4f_dict.py: to be edited to change the parameters of the VP1D4f computation (see below for a dictionary of parameters)

  • VP1D4f.py: contains the VP1D4f class and main functions (not to be edited)

  • VP1D4f_modules.py: contains the methods to run VP1D4f (not to be edited)

  • VP1D4f_AnalyzeData.m: MATLAB script to analyze the output saved in the .mat file

  • Once VP1D4f_dict.py has been edited with the relevant parameters, run the file as

  python3 VP1D4f.py

  or

  nohup python3 -u VP1D4f.py &>VP1D4f.out < /dev/null &

  The list of Python packages and their version are specified in modules_version.txt

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Parameter dictionary for VP1D4f

• kappa: double; value of κ defining the fluid reduction

• Tf: double; duration of the integration (in units of ω_p^-1)

• integrator_kinetic: string ('position-Verlet', 'velocity-Verlet', 'Forest-Ruth', 'PEFRL', 'BM4', 'BM6'); choice of solver for the integration of the Vlasov equation

  • 'Forest-Ruth' from Forest, Ruth, Physica D 43, 105 (1990)
  • 'PEFRL' from Omelyan, Mryglod, Folk, Comput. Phys. Commun. 146, 188 (2002)
  • 'BM4' and 'BM6' refer respectively to BM₆4 and BM₁₀6 from Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002)

• nsteps: integer; number of steps in one period of plasma oscillations (1/ω_p) for the integration of the Vlasov equation

• integrator_fluid: string ('RK45', ‘RK23’, ‘DOP853’, ‘BDF’, ‘LSODA’); choice of solver for the integration of the fluid equation (see ivp_solve for more details)

• precision: double; numerical precision of the integrator for the fluid equations; threshold for the Fourier transforms

• n_casimirs: integer; number of Casimir invariants to be monitored

• Lx: double; the x-axis is (-Lx, Lx)

• Lv: double; the v-axis is (-Lv, Lv)

• Nx: integer; number of points in x to represent the field variables

• Nv: integer; number of points in v to represent the field variables

• f_init: lambda function; initial distribution f(x,v,t=0)

• output_var: string in ['E', 'rho', 'u', 'P', 'q']; variable to be saved in a .mat file

• output_modes: integer or string in ['all', 'real']; number of Fourier modes of the variable to be saved in the .mat file; if 'all', all Fourier modes are saved; if 'real', the variable is saved in real space; use the MATLAB script VP1D4f_AnalyzeData.m to plot the output

• Kinetic: list of strings in ['Compute', 'Plot', 'Save']; list of instructions for the Vlasov-Poisson simulation; if contains 'Save', the results are saved in a .mat file

• Fluid: list of strings in ['Compute', 'Plot', 'Save']; list of instructions for the fluid simulation; if contains 'Save', the results are saved in a .mat file

• darkmode: boolean; if True, plots are done in dark mode

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Reference: C. Chandre, B.A. Shadwick, Four-field Hamiltonian fluid closures of the one-dimensional Vlasov-Poisson equation, Physics of Plasmas 29, 102101 (2022); arXiv:2206.06850

@article{chandre2022,
         title = {Four-field Hamiltonian fluid closures of the one-dimensional Vlasov-Poisson equation},
         author = {Chandre, C. and Shadwick, B.A.},
         journal = {Physics of Plasmas},
         volume = {29},
         number = {10},
         pages = {102101},
         year = {2022},
         doi = {10.1063/5.0102418},
         URL = {https://doi.org/10.1063/5.0102418}
}

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