Abstract
The goal of this paper is to develop a nonstandard finite difference-based numerical technique for solving the one-dimensional linear and non-linear Fokker–Planck equations. Characteristics of the nonstandard finite difference method are presented to understand the development of the proposed method. Conditions for the dynamic consistency of positivity and stability of the schemes are obtained. Numerical experiments have been carried out to demonstrate the competitiveness of the proposed methods in comparison to some existing standard methods. In support of the proposed method and analysis, the \(l_2\) and \(l_\infty \) errors are also presented.
Similar content being viewed by others
References
Risken H (1996) The Fokker–Planck equation. Springer, Berlin, pp 63–95
Zorzano MP, Mais H, Vázquez L (1998) Numerical solution for Fokker–Planck equations in accelerators. Physica D 113(2–4):379–381
Choe HJ, Ahn C, Kim BJ, Ma Y-K (2013) Copulas from the Fokker–Planck equation. J Math Anal Appl 406(2):519–530
Kopp A, Büsching I, Strauss RD, Potgieter MS (2012) A stochastic differential equation code for multidimensional Fokker–Planck type problems. Comput Phys Commun 183(3):530–542
Risken H, Eberly JH (1985) The Fokker-Planck equation, methods of solution and applications. J Opt Soc Am B 2(3):508
Zorzano MP, Mais H, Vazquez L (1999) Numerical solution of two dimensional Fokker–Planck equations. Appl Math Comput 98(2–3):109–117
Wojtkiewicz SF, Bergman, LA (2000) Numerical solution of high dimensional Fokker–Planck equations. In: 8th ASCE specialty conference on probablistic mechanics and structural reliability, Notre Dame, IN, USA. Citeseer, 2000
Liu S, Li W, Zha H, Zhou H (2022) Neural parametric Fokker–Planck equation. SIAM J Numer Anal 60(3):1385–1449
Palleschi V, De Rosa M (1992) Numerical solution of the Fokker–Planck equation. II. Multidimensional case. Phys Lett A 163(5–6):381–391
Dehghan M, Tatari M (2006) The use of He’s variational iteration method for solving a Fokker–Planck equation. Phys Scr 74(3):310
Tatari M, Dehghan M, Razzaghi M (2007) Application of the Adomian decomposition method for the Fokker–Planck equation. Math Comput Model 45(5–6):639–650
Jafari MA, Aminataei A (2009) Application of homotopy perturbation method in the solution of Fokker–Planck equation. Phys Scr 80(5):055001
Lakestani M, Dehghan M (2009) Numerical solution of Fokker–Planck equation using the cubic b-spline scaling functions. Numeri Methods Partial Differ Equ 25(2):418–429
Kazem S, Rad JA, Parand K (2012) Radial basis functions methods for solving Fokker–Planck equation. Eng Anal Boundary Elem 36(2):181–189
Kumar M, Pandit S (2015) An efficient algorithm based on Haar wavelets for numerical simulation of Fokker–Planck equations with constants and variable coefficients. Int J Numer Methods Heat Fluid Flow 25:41–56
Zanella M (2020) Structure preserving stochastic Galerkin methods for Fokker–Planck equations with background interactions. Math Comput Simul 168:28–47
Srinivasa K, Rezazadeh H, Adel W (2022) An effective numerical simulation for solving a class of Fokker–Planck equations using Laguerre wavelet method. Math Methods Appl Sci 45(11):6824–6843
Wang T, Chai G (2022) A fully discrete pseudospectral method for the nonlinear Fokker–Planck equations on the whole line. Appl Numer Math 174:17–33
Zhang Y, Yuen K-V (2022) Physically guided deep learning solver for time-dependent Fokker–Planck equation. Int J Non-Linear Mech 147:104202
Xu Y, Zhang H, Li Y, Zhou K, Liu Q, Kurths J (2020) Solving Fokker–Planck equation using deep learning. Chaos 30(1):013133
Zhang H, Xu Y, Liu Q, Wang X, Li Y (2022) Solving Fokker–Planck equations using deep KD-tree with a small amount of data. Nonlinear Dyn 1–15
Brunken J, Smetana K (2022) Stable and efficient Petrov–Galerkin methods for a kinetic Fokker–Planck equation. SIAM J Numer Anal 60(1):157–179
Tang K, Wan X, Liao Q (2022) Adaptive deep density approximation for Fokker–Planck equations. J Comput Phys 457:111080
Pareschi L, Zanella M (2018) Structure preserving schemes for nonlinear Fokker–Planck equations and applications. J Sci Comput 74(3):1575–1600
Duan C, Chen W, Liu C, Wang C, Zhou S (2022) Convergence analysis of structure-preserving numerical methods for nonlinear Fokker–Planck equations with nonlocal interactions. Math Methods Appl Sci 45(7):3764–3781
Harrison GW (1988) Numerical solution of the Fokker–Planck equation using moving finite elements. Numer Methods Partial Differ Equ 4(3):219–232
Qian Y, Wang Z, Zhou S (2019) A conservative, free energy dissipating, and positivity preserving finite difference scheme for multi-dimensional nonlocal Fokker–Planck equation. J Comput Phys 386:22–36
Mickens RE (1994) Nonstandard finite difference models of differential equations. World Scientific, Singapore
Anguelov R, Lubuma JM-S (2001) Contributions to the mathematics of the nonstandard finite difference method and applications. Numer Methods Partial Differ Equ 17(5):518–543
Mickens RE (1999) An introduction to nonstandard finite difference schemes. J Comput Acoust 7(01):39–58
Mickens RE (2007) Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition. Numer Methods Partial Differ Equ 23(3):672–691
Lubuma JM-S, Patidar KC (2006) Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems. J Comput Appl Math 191(2):228–238
Patidar KC, Sharma KK (2006) \(\varepsilon \)-uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with small delay. Appl Math Comput 175(1):864–890
Patidar KC, Sharma KK (2006) Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance. Int J Numer Methods Eng 66(2):272–296
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Neena, A.S., Clemence-Mkhope, D.P. & Awasthi, A. Nonstandard finite difference schemes for linear and non-linear Fokker–Planck equations. J Eng Math 145, 11 (2024). https://doi.org/10.1007/s10665-024-10346-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10665-024-10346-2