A semianalytic meshless approach to the Fokker-Planck equation with application to hybrid systems

Abstract

A semianalytic partition of unity finite element method (PUFEM) is presented to solve the transient Fokker-Planck equation (FPE) for high-dimensional nonlinear dynamical systems. Meshless spatial discretization of the PUFEM is employed to develop linear ordinary differential equations for the time varying coefficients of the local shape functions. It is shown that a similarity transformation to the modal coordinates brings out numerous spurious modes in the eigenspace of the discretized FPE operator. The identification and elimination of these modes leads to an analytical solution of the ODEs for the coefficients of the shape functions developed by the spatial discretization. Furthermore, the retention of only a few admissible modes leads to a significant order-reduction in the transient problem. Equation-error bounds for the approximation are presented and it is shown that the equation-error at the initial time resulting from the retained set of admissible modes is an upper bound for all subsequent times. Results are presented for a two-dimensional nonlinear stochastic oscillator. Some initial ideas are presented for the application of this approach to determining the probabilistic behavior of stochastic hybrid dynamical systems with Markovian switching.