Efficient Solution of Fokker–Planck Equations in Two Dimensions

Donald Michael McFarland, Fei Ye, Chao Zong, Rui Zhu, Tao Han, Hangyu Fu, Lawrence A. Bergman and Huancai Lu ()
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Donald Michael McFarland: Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China
Fei Ye: Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China
Chao Zong: Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China
Rui Zhu: Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China
Tao Han: Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo 315201, China
Hangyu Fu: Sound and Vibration Laboratory, College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, China
Lawrence A. Bergman: Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Huancai Lu: Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo 315201, China

Mathematics, 2025, vol. 13, issue 3, 1-20

Abstract: 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.

Keywords: Fokker–Planck equation; finite element analysis; dimension reduction; operator splitting (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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