A nodal integral method for the Fokker-Planck equation

The Fokker-Planck equation is important in the kinetic theory of plasmas for the description of long-range coulomb collisions of charged particles. Hence, it is used extensively in modeling fusion devices, such as magnetic mirrors and certain aspects of tokamaks. The authors have developed a nodal integral method (NIM) for the accurate numerical solution of the Fokker-Planck equation, applied it to test problems, and compared the results obtained with those obtained using a finite difference method (FDM). These comparisons show that the NIM is more accurate and more computationally efficient than the FDM, especially in the calculation of particle and energy leakages and when applied to more difficult test problems. The new method significantly extends ideas developed previously to more complicated partial differential equations (PDEs) in two important ways. Since the nonlinearities in the Fokker-Planck equation are considerably more complicated than those that arise in the Navier-Stokes equations and the Boussinesq equations, the NIM developed here extends the general technique farther into the nonlinear regime. Further, since the Fokker-Planck equation is singular at the origin in spherical velocity coordinates, the geometry relevant to most practical problems, special origin equations had to be developed for the computational elements adjacent to the v = 0 boundary