| Contents |
| The Fokker-Planck-Landau(FPL) equation is a kinetic model for the time evolution of a plasma.
Our focus at present is on the numerical approximation of the Landau collision operator hence we
consider the spatially homogeneous equation. This equation is
\beq \frac{\partial f}{\partial t} = q\Phi(f,f),\;\; f(v,0) = f_0(v), \eeq
where $v=(v_1,v_2,v_3) \in R^3,\; f(v,t)$ is the distribution function for a single charged species,
and $\Phi(f,f)$ is the Landau collision operator for Coulomb collisions.
The equation (1) is put into a form so that finite difference methods for parabolic type PDE's can be
applied. Through a change of independent variable the equation is reformulated so that finite
differencing in velocity space is done on a bounded domain with well defined boundary conditions. The
reformulate FPL equation is approximated by a semi-implicit difference equation on a uniform grid and
is solved by the SOR algorithm. The coefficients in the difference equation are obtained as derivatives
of a Rosenbluth potential. By taking advantage of regularity properties of the solution (which are assumed
to exist although a formal proof is lacking) the time needed to compute the Rosenbluth potential is reduced
and accuracy is maintained by doing numerical integrations on courser subgrids of the finite difference grid.
Also parallel programming is easily applied to further reduce computation time. Some analysis is carried
out on the stability and accuracy of the numerical method. A number of computational examples are given
showing the convergence of the solution to the steady state. |
|