The quadrature discretization method in the solution of the Fokker-Planck equation with nonclassical basis functions

Fokker-Planck equations are used extensively to study a variety of problems in nonequilibrium statistical mechanics. A discretization method referred to as the quadrature discretization method (QDM) is introduced for the time-dependent solution of Fokker-Planck equations. The QDM is based on the discretization of the probability density function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to some weight function. For the Fokker-Planck equation, the weight functions that have often provided rapid convergence of the eigenvalues of the Fokker-Planck operator are the steady distributions at infinite time. Calculations are carried out for several systems with bistable potentials that arise in the study of optical bistability. reactive systems and climate models. The rate of convergence of the eigenvalues and the eigenfunctions of the Fokker-Planck equation is very rapid with this approach. The time evolution is determined in terms of the expansion of the distribution function in the eigenfunctions. (C) 1997 American Institute of Physics.