Abstract
A practical, implicit finite-difference scheme, accurate to second order both in time step and mesh size, is presented to solve nonlinear Fokker–Planck (FP) equations in nonuniform grids. Weighting coefficients are introduced for proper equilibration, and the conditions under which the nonuniformity correction is essential in them to ensure physically meaningful solutions are analyzed. Conservation of particle number and energy is addressed, as well as the constraints on the design of nonuniform grids. The proposed scheme is shown to efficiently handle both linear and weakly nonlinear problems and, in addition, its ability to provide solutions to stronger nonlinear situations is demonstrated. Examples of FP equations (describing Coulomb collisions, Compton scattering, and rf heating and current drive) are solved to illustrate the effectiveness of nonuniform grids in reducing computational effort with no significant deterioration in accuracy, and in increasing the latter at no expense of the former.
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