SOLUTION OF NONLINEAR FOKKER-PLANCK EQUATIONS

A finite-difference method for solving a general class of linear and n onlinear time-dependent Fokker-Planck equations, which is based on a K -point Stirling interpolation formula, is suggested. It has a fifth-or der convergence in time and a 2 Kth-order convergence in space and all ows one to achieve a given level of accuracy with a slow (or even with out) increase in the number of grid points. The most appealing feature s of the method are perhaps that it is norm conserved, and equilibrium preserving in the sense that every equilibrium solution of the analyt ic equations is also an equilibrium solution of the discretized equati ons. The method is applied to a nonlinear stochastic mean-field model introduced by Kometani and Shimizu [J. Stat. Phys. 13, 473 (1983)], wh ich exhibits a phase transition. The results are compared with those o btained with other methods that rely on not too well controlled approx imations. Our finite-difference scheme permits us to establish the reg ion of validity and the limitations of those approximations. The nonli nearity of the system is found to be an obstacle for the application o f Suzuki's scaling ideas, which are known to be suitable for linear pr oblems. But what is most remarkable is that this nonlinearity allows f or transient bimodality in a globally monostable case, even though the re is no ''flat'' region in the potential.