This work presents an effective numerical technique to solve linear and nonlinear Fokker–Planck equations (FPE) with constant or variable drift and diffusion coefficients. The linear FPE is first semi-discretized by the method of lines (MOL) in the x direction, resulting in a set of ordinary differential equations (ODEs) in the temporal direction. For the nonlinear FPE, the discretized equation resulting from the temporal discretization is linearized using Taylor series expansion. The subsequent system of ODEs is discretized using backward differentiation formulas (BDFs) of different orders in the time direction. The local truncation errors of the proposed schemes with first and second-order BDFs are shown to be the same as those of the Chang–Cooper scheme (Chang and Cooper in J Comput Phys 6(1):1–16, 1970) with first-order and second-order time differencing, respectively. Convergence analysis of the BDF schemes is presented along with the existence and uniqueness as well as the Lyapunov stability results. Numerical errors are calculated using <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12046_2025_2871_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12046_2025_2871_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$L_{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>, and RMS norms, and the validation of the proposed schemes is done by comparing the numerical results with the analytic solutions. The reliability and accuracy of the BDF schemes are presented through the comparison of the results with existing techniques in the literature.

Numerical method for Fokker–Planck equations based on backward differentiation formulas

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发表日期：

2025年11月18日

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A S Neena,
Dominic P Clemence-Mkhope,
Ashish Awasthi

摘要

This work presents an effective numerical technique to solve linear and nonlinear Fokker–Planck equations (FPE) with constant or variable drift and diffusion coefficients. The linear FPE is first semi-discretized by the method of lines (MOL) in the x direction, resulting in a set of ordinary differential equations (ODEs) in the temporal direction. For the nonlinear FPE, the discretized equation resulting from the temporal discretization is linearized using Taylor series expansion. The subsequent system of ODEs is discretized using backward differentiation formulas (BDFs) of different orders in the time direction. The local truncation errors of the proposed schemes with first and second-order BDFs are shown to be the same as those of the Chang–Cooper scheme (Chang and Cooper in J Comput Phys 6(1):1–16, 1970) with first-order and second-order time differencing, respectively. Convergence analysis of the BDF schemes is presented along with the existence and uniqueness as well as the Lyapunov stability results. Numerical errors are calculated using $L_2$ $L_{2}$ , $L_{\infty }$ $L_{\infty}$ , and RMS norms, and the validation of the proposed schemes is done by comparing the numerical results with the analytic solutions. The reliability and accuracy of the BDF schemes are presented through the comparison of the results with existing techniques in the literature.