An Implicit Energy-Conservative 2D Fokker–Planck Algorithm: II. Jacobian-Free Newton–Krylov Solver

Energy-conservative implicit integration schemes for the Fokker–Planck transport equation in multidimensional geometries require inverting a dense, non-symmetric matrix (Jacobian), which is very expensive to store and solve using standard solvers. However, these limitations can be overcome with Newton–Krylov iterative techniques, since they can be implemented Jacobian-free (the Jacobian matrix from Newtonʹs algorithm is never formed nor stored to proceed with the iteration), and their convergence can be accelerated by preconditioning the original problem. In this document, the efficient numerical implementation of an implicit energy-conservative scheme for multidimensional Fokker–Planck problems using multigrid-preconditioned Krylov methods is discussed. Results show that multigrid preconditioning is very effective in speeding convergence and decreasing CPU requirements, particularly in fine meshes. The solver is demonstrated on grids up to 128×128 points in a 2D cylindrical velocity space (vr, vp) with implicit time steps of the order of the collisional time scale of the problem, τ. The method preserves particles exactly, and energy conservation is improved over alternative approaches, particularly in coarse meshes. Typical errors in the total energy over a time period of 10τ remain below a percent.