Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems

Abstract

The response of a structural system to white noise excitation (deltacorrelated) constitutes a Markov vector process whose transitional probability density function (TPDF) is governed by both the forward Fokker-Planck and backward Kolmogorov equations. Numerical solution of these equations by finite element and finite difference methods for dynamical systems of engineering interest has been hindered by the problem of dimensionality. In this paper numerical solution of the stationary and transient form of the Fokker-Planck (FP) equation corresponding to two state nonlinear systems is obtained by standard sequential finite element method (FEM) using C 0 shape function and Crank-Nicholson time integration scheme. The method is applied to Van-der-Pol and Duffing oscillators providing good agreement between results obtained by it and exact results. An extension of the finite difference discretization scheme developed by Spencer, Bergman and Wojtkiewicz is also presented. This paper presents an extension of the finite difference method for the solution of FP equation up to four dimensions. The difficulties associated in extending these methods to higher dimensional systems are discussed.

Calculation of high-dimensional probability density functions of stochastically excited nonlinear mechanical systems

“…Problems with stationary pdfs that are not too far from Gaussian can be solved for significantly higher dimensions than with comparable numerical methods. In the case of nonlinear systems with stationary pdfs that diverge strongly from Gaussian distributions (such as considered in [3], for example), there are obvious limitations to the method as outlined in this paper. However, the use of appropriate nonlinear coordinate transformations, as described in [10] or [11], as well as the use of well-adapted weighting functions also allows the efficient solution of problems with distributions far from Gaussian.…”

“…Methods for the discretization of the state space include Finite Element and Finite Difference methods, such as in [2] and [3], but these methods bear the disadvantage that, generally, a d-dimensional infinite space has to be handled for the pdf. Semi-analytical methods include the path integral method (see for example [1,4,5]) and the cell mapping method [6].…”

Variational superposed Gaussian approximation for time-dependent solutions of Langevin equations

“…[21] and the references therein). M ore-direct num erical schem es, such as a finite-element m ethod [22,23] and a finite-difference m ethod [24], have also been studied. These approaches, however, have high com pu tational costs and hence are not suited for tim e-dependent solutions.…”

Discontinuous Galerkin Methods for High-Dimensional Fokker-Planck Equations in Stochastic Dynamics

“…by [17], where domain decomposition for higher dimensions is suggested and shown for 2D and 3D. A comparison between Finite Element and Finite Difference methods can be found in [5] using the example of up to 4D non-linear oscillators. To approach the solution of higher dimensional problems, multi-scale Finite Element methods are suggested e.g.…”