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https://github.com/cchandre/vlasov1d

One-dimensional Vlasov-Poisson equation and its Hamiltonian fluid reductions
https://github.com/cchandre/vlasov1d

fluid-simulation fluid-solver hamiltonian hamiltonian-dynamics livescript mathematica matlab matlab-code numpy plasma-physics python3 scipy vlasov-equation

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

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



- [`Vlasov1D_4field.mlx`](https://github.com/cchandre/Vlasov1D/blob/main/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`](https://github.com/cchandre/Vlasov1D/blob/main/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`](https://github.com/cchandre/Vlasov1D/blob/main/VP1D4f_dict.py): to be edited to change the parameters of the VP1D4f computation (see below for a dictionary of parameters)



  - [`VP1D4f.py`](https://github.com/cchandre/Vlasov1D/blob/main/VP1D4f.py): contains the VP1D4f class and main functions (not to be edited)



  - [`VP1D4f_modules.py`](https://github.com/cchandre/Vlasov1D/blob/main/VP1D4f_modules.py): contains the methods to run VP1D4f (not to be edited)

  

  - [`VP1D4f_AnalyzeData.m`](https://github.com/cchandre/Vlasov1D/blob/main/VP1D4f_AnalyzeData.m): MATLAB script to analyze the output saved in the `.mat` file  



  - Once [`VP1D4f_dict.py`](https://github.com/cchandre/Vlasov1D/blob/main/VP1D4f_dict.py) has been edited with the relevant parameters, run the file as 

  ```sh

  python3 VP1D4f.py

  ```

  or 

  ```sh

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

  ```

  The list of Python packages and their version are specified in [`modules_version.txt`](https://github.com/cchandre/Vlasov1D/blob/main/modules_version.txt)

___

##  Parameter dictionary for VP1D4f



- *kappa*: double; value of κ defining the fluid reduction 

- *Tf*: double; duration of the integration (in units of *ωp-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)](https://doi.org/10.1016/0167-2789(90)90019-L)

   - 'PEFRL' from [Omelyan, Mryglod, Folk, Comput. Phys. Commun. 146, 188 (2002)](https://doi.org/10.1016/S0010-4655(02)00451-4)

   - 'BM4' and 'BM6' refer respectively to BM64 and BM106 from [Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002)](https://doi.org/10.1016/S0377-0427(01)00492-7)

- *nsteps*: integer; number of steps in one period of plasma oscillations (1/*ωp*) 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](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html) 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`](https://github.com/cchandre/Vlasov1D/blob/main/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



---

Reference: C. Chandre, B.A. Shadwick, *Four-field Hamiltonian fluid closures of the one-dimensional Vlasov-Poisson equation*, [Physics of Plasmas](https://aip.scitation.org/journal/php) 29, 102101 (2022); [arXiv:2206.06850](https://arxiv.org/abs/2206.06850)



```bibtex

@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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