Solution of nonlinear Fokker-Planck equations

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Abstract
A finite-difference method for solving a general class of linear and nonlinear time-dependent Fokker-Planck equations, which is based on a K-point Stirling interpolation formula, is suggested. It has a fifth-order convergence in time and a 2Kth-order convergence in space and allows one to achieve a given level of accuracy with a slow (or even without) increase in the number of grid points. The most appealing features of the method are perhaps that it is norm conserved, and equilibrium preserving in the sense that every equilibrium solution of the analytic equations is also an equilibrium solution of the discretized equations. The method is applied to a nonlinear stochastic mean-field model introduced by Kometani and Shimizu [J. Stat. Phys. 13, 473 (1983)], which exhibits a phase transition. The results are compared with those obtained with other methods that rely on not too well controlled approximations. Our finite-difference scheme permits us to establish the region of validity and the limitations of those approximations. The nonlinearity of the system is found to be an obstacle for the application of Suzuki’s scaling ideas, which are known to be suitable for linear problems. But what is most remarkable is that this nonlinearity allows for transient bimodality in a globally monostable case, even though there is no ‘‘flat’’ region in the potential. © 1996 The American Physical Society.