Efficient Solution of Fokker–Planck Equations in Two Dimensions

Finite element analysis (FEA) of the Fokker–Planck equation governing the nonstationary joint probability density function of the responses of a dynamical system produces a large set of ordinary differential equations, and computations become impractical for systems with as few as four states. Nonetheless, FEA remains of interest for small systems—for example, for the generation of baseline performance data and reference solutions for the evaluation of machine learning-based methods. We examine the effectiveness of two techniques which, while they are well established, have not to our knowledge been applied to this problem previously: reduction of the equations onto a smaller basis comprising selected eigenvectors of one of the coefficient matrices, and splitting of the other coefficient matrix. The reduction was only moderately effective, requiring a much larger basis than was expected and producing solutions with clear artifacts. Operator splitting, however, performed very well. While the methods can be combined, our results indicate that splitting alone is an effective and generally preferable approach.