Numerical solution of the Fokker-Planck-Landau equation by fast spectral methods

Kinetic density

Nowadays, numerical simulations of plasmas are receiving a great deal of attention both in research and in industry thanks to the numerous applications directly connected to these phenomena. In addition, there exist many practical situations in which the so-called Coulomb collisions are fundamental for correctly describing the plasma dynamics as for instance in magnetic fusion devices (like tokamak devices). The Fokker-Planck-Landau (FPL) equation is used to describe the binary collisions between charged particles in plasma physics.

A new approach for the accurate numerical solution of the FPL equation has been presented recently in [1, 2]. The method is based on a fast spectral solver for the efficient solution of the collision operator. The use of a suitable explicit Runge-Kutta solver for the time integration of the collision phase permits to avoid excessive small time steps induced by the stiffness of the diffusive collision operator. Here we present some numerical simulation of the relaxation process in a three-dimensional Coulomb gas (see [3,4]). We refer also to [5] for the development of Monte Carlo methods for the Landau-Fokker-Planck equation.

The relaxation process in 3D

We consider the relaxation process for a three dimensional Coulomb gas (a=-3). The initial data is chosen as the sum of two symmetric Maxwellian functions

f₀(v) = 5[exp(-(|v-v₁|²)/(2v_t²)) + exp(-(|v-v₂|²)/(2v_t²))]/(2(2pv_t²)^3/2),

with v₁ = (1.25,1.25,0),  v₂ = (-1.25,-1.25,0) and the thermal velocity is v_t=0.4. The final time of the simulation is T = 80. The movie reports the time evolution of the level set of the distribution function f(t,v_x,v_y,v_z)=0.02 obtained with n=32 modes. Initially the level set of the initial data corresponds to two ``spheres'' in the velocity space. Then, the two distributions start to merge until the stationary state characterized by a Maxwellian distribution with zero mean velocity is reached. This is represented by a single centered sphere.

References

1. L.Pareschi, G.Russo, G.Toscani, Methode spectrale rapide pour l'equation de Fokker Planck Landau, C. R. Acad. Sci. Paris, t.330, Serie I, pp.517-522, (2000). 
2. L.Pareschi, G.Russo, G.Toscani, Fast spectral methods for the Fokker-Planck-Landau collision operator, J. Comp. Phys. 165, pp. 216-236, (2000). 
3. F.Filbet, L.Pareschi, A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the non homogeneous case, Journal of Computational Physics, 179, 1-26 (2002). 
4. F.Filbet, L.Pareschi, Numerical solution of the Fokker-Planck-Landau equation by spectral methods. Commun. Math. Sci. 1 (2003), no. 1, 206--207. 
5. Caflisch R., Dimarco G., Pareschi L., Direct simulation Monte Carlo schemes for Coulomb interactions in plasmas, Comm. App. Ind. Math., 1, (2010), pp. 72-91.