Cell renormalized FPK equation for stochastic non-linear systems

摘要

The Fokker-Planck-Kolmogorov (FPK) equation, as a well investigated partial differential equation, is of great significance to stochastic dynamics due to its theoretical rigor and exactness. However, practical difficulties with the FPK method are encountered when analysis of multi-degree-of-freedom (MDOF) systems with arbitrary nonlinearity is required. In the present paper, a cell renormalized method (CRM) which is based on a numerical determination of response statistical moments of the FPK equation is developed. Specifically, by invoking the concept of equivalence of probability flux, a cell renormalization procedure and a reconstruction scheme of derivative moments are introduced to divide the continuous state space into a discretized region of cells so that numerical derivative moments is derived. Subsequently, the Cell Renormalized FPK (CR-FPK) equation can be solved by a finite difference algorithm. Two numerical examples are included, and the effectiveness of the proposed method is assessed.