Guiding centre (X,Y) simulation

Equations

The guiding_centre executable deals with the following model:

\[ \partial_t f(t, x, y) + (E(t, x, y)\wedge e_z) \cdot \nabla f(t, x, y) = 0, \\ - \Delta \phi(t, x, y) = -\partial_x^2 \phi(t, x, y) -\partial_y^2 \phi(t, x, y) = f(t, x, y), \\ E(t, x, y) = - \nabla \phi(t, x, y). \]

Remark: the advection field of the advection equation is given by \(\frac{E\wedge B}{|B|^2} = E(t, x, y)\wedge e_z\) with \(B\) the magnetic field supposed of norm 1, (and \(e_z\) a vector perpendicular to the \((x,y)\) plane in the direct direction).

Operators

The simulations uses the following operators:

  • advection equation: BslAdvection1D operator with a Strang splitting along \(x\) and \(y\). The time integration methods applied to solve the characteristic equation are explicit Euler methods;
  • Poisson equation: FFTPoissonSolver solver using FFT to solve the Poisson equation on a periodic domain (and compute the electric field);
  • equations coupling: PredCorrRK2XY using a RK2 time integration method.

Execution space

The simulation runs on GPU/CPU.

Output files

The guiding_centre executable creates in the working folder an output/ folder containing the output files: density, electrostatic potential and electric field values on the grid for some regular time steps.

Command line

./guiding_centre_XY <path_to_params.yaml>

Simulation

Grid

The simulation runs on a grid on \([0, 4\pi]\times[0, 2\pi]\) with 64 uniformly distributed points in each direction.

Test case: Kelvin-Helmholtz instability test case

The chosen initial conditions are

\[ f(0, x, y) = f_{\text{eq}}(x,y) + \varepsilon\cos(kx), \\ f_{\text{eq}}(x,y) = \sin(y) \]

with \(\varepsilon = 0.015\) the amplitude of perturbation and \(k = 2\pi/ L_x = 0.5\) the mode of the perturbation.

See more in initialisations.

Results treatments - diagnosis

In post-process/PythonScripts/geometryXY/ folder, python files are available to plot the results saved in the output/ folder:

  • animation.py displays and can save the curves of the density \(f\) and the electrostatic potential \(\phi\) for each time steps saved in the output/ folder.

The command line is given by

python3 animation.py --name=<name_file_to_save> --folder=<path_to_output>
  • energy_conservation.py plots the evolution of the energy of the system along the simulation:
\[ d\mathcal{M}: t\mapsto \int_{\Omega} |E(t)|^2 - \int_{\Omega} |E_0|^2 \]

The command line is given by

python3 energy_conservation.py --name=<name_file_to_save> --folder=<path_to_output>
  • mass_conservation.py plots the evolution of the mass of the system along the simulation:
\[ d\mathcal{W}: t\mapsto \int_{\Omega} f(t) - \int_{\Omega} f_0 \]

The command line is given by

python3 mass_conservation.py --name=<name_file_to_save> --folder=<path_to_output>
  • plot_individual_curve.py plots the curves of the density \(f\) and the electrostatic potential \(\phi\) for a given time

The command line is given by

python3 plot_individual_curve.py --name=<name_file_to_save> --time=<given_time> --folder=<path_to_output>
  • plot_L2_norms.py plots the \(\mathcal{L}^2\) norms of the density \(f - f_{eq}\) and the electrostatic potential \(\phi\) along the simulation

The command line is given by

python3 plot_L2_norms.py --name=<name_file_to_save> --folder=<path_to_output>

Contents

  • guiding_centre.cpp: executable of a guiding-centre equation on \((x,y)\) geometry with Kelvin-Helmholtz instability test case initial conditions.
  • params.yaml: contains the parameters of the simulation. It needs to be added at the command line to launch the executable.

References

[1] Eric Sonnendrücker, Jean Roche, Pierre Bertrand and Alain Ghizzo. "The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation", on Journal of Computational Physics, Oct. 1998.

[2] Nicolas Crouseilles, Michel Mehrenberger, Eric Sonnendrücker. "Conservative semi-Lagrangian schemes for Vlasov equations". Oct. 2009.