An implicit energy-conservative 2D Fokker-Planck algorithm - II. Jacobian-free Newton-Krylov solver

Energy-conservative implicit integration schemes for the Fokker-Planck tran sport equation in multidimensional geometries require inverting a dense, no n-symmetric matrix (Jacobian), which is very expensive to store and solve u sing standard solvers. However, these limitations can be overcome with Newt on-Krylov iterative techniques, since they can be implemented Jacobian-free (the Jacobian matrix from Newton's algorithm is never formed nor stored to proceed with the iteration), and their convergence can be accelerated by p reconditioning the original problem, In this document, the efficient numeri cal implementation of an implicit energy-conservative scheme for multidimen sional Fokker-Planck problems using multigrid-preconditioned Krylov methods is discussed, Results show that multigrid preconditioning is very effectiv e in speeding convergence and decreasing CPU requirements, particularly in fine meshes. The solver is demonstrated on grids up to 128 x 128 points in a 2D cylindrical velocity space (upsilon(r), upsilon(p)) with implicit time steps of the order of the collisional time scale of the problem, tau. The method preserves particles exactly, and energy conservation is improved ove r alternative approaches, particularly in coarse meshes, Typical errors in the total energy over a rime period of 10 tau remain below a percent. (C) 2 000 Academic Press.