Kinetic Software

Kinetic Codes to Solve Fokker-Planck Equation

At the most fundamental level a plasma can be described by kinetic equations for the distribution function of each particle species in the six-dimensional phase-space and time. These kinetic equations are coupled to each other through distribution function and self-consistent electric and magnetic fields.

For a wide class of problems it is possible to reduce detailed and often cumbersome description of plasma down to a much more manageable formulation, in which Fokker-Planck (FP) equations play a fundamental role. However, excluding a few particular cases, the solutions to such FP equations are not available in an analytically closed form. So numerical and approximate solutions are the only resource when proper modelling is required. Therefore, the ability to efficiently provide accurate and stable numerical solutions to general linear or non-linear FP equations establishes itself as a key factor in tokamak plasma modelling. Using FP equations is especially essential when modelling Coulomb collisional slowing down, transport and general wave-plasma interactions, resulting either in wave damping and energy absorption or in wave scattering.

Since the numerical solution of FP equations in most applications is only a part of a more complex modelling scheme, the major goal of any solving method is to reduce as much as possible the cost in computing effort without compromising in accuracy or stability. As the computational cost relates directly to the overall size of the mesh used to obtain a discrete solution of the FP equation, one obvious way to reduce such cost is to employ non-uniform meshes, distributing the mesh points location according to the real needs of the problem being addressed. The guidance for this rearrangement should be established by minimising the errors introduced when turning a continuous FP equation into a series of discrete relations, increasing the accuracy for a given number of mesh points or, conversely, reducing the mesh size whilst maintaining the accuracy level.

Within the most widespread approaches for solving FP equations numerically, some finite-difference schemes may be singled out, which are intrinsically particle conserving and are able to preserve the non-negative character of their solutions, as well as exact representations of the equilibrium state. When dealing with non-linear problems, this type of schemes may be extended to non-uniform grids, not only by means of the proper coefficients weighting, but also by redefining the concept of quasi-equilibrium solutions. Indeed, this is the key factor in order to maintain a rather general approach when solving FP equations, without resorting to particular properties of certain FP operators as, for instance, their ability to be rewritten in a certain special form.

The above considerations were successfully applied to the solution of two representative non-linear problems in plasma physics, that is, the Coulomb scattering of a like-charged particle population and the Compton scattering of photons, in frequency space, due to the interaction with an electronic population in thermal equilibrium. The weakly non-linear nature of the first problem is related with the integral form of the friction and diffusion coefficients, which in turn makes the computational effort to scale roughly as N in 2.4 degree, with N being the number of cells in the mesh. The second is strongly non-linear and represents a severe test to the proposed schemes.

Since the Coulomb scattering problem is energy conservative by definition, one suitable process to evaluate the numerical solution quality is to compute the magnitude of energy dissipation effects, introduced by the discretization procedure. It may be shown that such energy dissipation effects are quantified by an integral functional involving the mesh size function Δ(x) and some of its derivatives, the minimisation of such functions being used as a guide in designing suitable non-uniform grids. In fact, even a rough design consisting in the juxtaposing of two uniform grids of different mesh sizes (a finer one for the low velocity zone and a larger one for the high velocity zone), is able to reduce energy dissipation when compared with uniform grids of the same global size.