Finite difference approximations and dynamics simulations for the Lévy Fractional Klein‐Kramers equation

The Klein‐Kramers equation describes position and velocity distribution of Langevin dynamics, the diffusion equation and Fokker‐Planck equation are its special cases for characterizing position distribution and velocity distribution, respectively. Incorporating the mechanisms of Lévy flights into the Klein‐Kramers formalism leads to the Lévy fractional Klein‐Kramers equation, which can effectively describe Lévy flights in the presence of an external force field in the phase space. For numerically solving the Lévy fractional Klein‐Kramers equation, this article presents the explicit and implicit finite difference schemes. The discrete maximum principle is generalized, using this result the detailed stability and convergence analyses of the schemes are given. And the extrapolation and some other possible techniques for improving the convergent rate or making the schemes efficient in more general cases are also discussed. The extensive numerical experiments are performed to confirm the effectiveness of the numerical schemes or simulate the superdiffusion processes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011