We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky weight functions on the edges. This family particularly includes the well-established Scharfetter–Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection the choice of the optimal representative of the discretization family is highly non-trivial, while for large gradients the Scharfetter–Gummel scheme stands out compared to the others.
Keywords: Finite volume, Fokker–Planck, Stolarsky mean
@article{M2AN_2021__55_6_3017_0,
author = {Heida, Martin and Kantner, Markus and Stephan, Artur},
title = {Consistency and convergence for a family of finite volume discretizations of the {Fokker{\textendash}Planck} operator},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {3017--3042},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {6},
doi = {10.1051/m2an/2021078},
mrnumber = {4353557},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021078/}
}
TY - JOUR AU - Heida, Martin AU - Kantner, Markus AU - Stephan, Artur TI - Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 3017 EP - 3042 VL - 55 IS - 6 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021078/ DO - 10.1051/m2an/2021078 LA - en ID - M2AN_2021__55_6_3017_0 ER -
%0 Journal Article %A Heida, Martin %A Kantner, Markus %A Stephan, Artur %T Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 3017-3042 %V 55 %N 6 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021078/ %R 10.1051/m2an/2021078 %G en %F M2AN_2021__55_6_3017_0
Heida, Martin; Kantner, Markus; Stephan, Artur. Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 6, pp. 3017-3042. doi: 10.1051/m2an/2021078
[1] and , Relaxation methods applied to determine the motion in two dimensions of a viscous fluid past a fixed cylinder. Q. J. Mech. Appl. Math. 8 (1955) 129–145. | MR | Zbl | DOI
[2] , and , The finite volume scharfetter-gummel method for steady convection diffusion equations. Comput. Visulaization Sci. 1 (1998) 123–136. | Zbl | DOI
[3] , An algorithm with guaranteed convergence for finding a zero of a function. Comput. J. 14 (1971) 422–425. | MR | Zbl | DOI
[4] , and , Numerical simulation of semiconductor devices. Comput. Methods Appl. Mech. Eng. 75 (1989) 493–514. | MR | Zbl | DOI
[5] and , Convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations. Math. Comput. 85 (2016) 549–580. | MR | DOI
[6] and , Numerical analysis of a robust free energy diminishing finite volume scheme for parabolic equations with gradient structure. Found. Comput. Math. 17 (2017) 1525–1584. | MR | DOI
[7] and , Finite-volume schemes for noncoercive elliptic problems with neumann boundary conditions. IMA J. Numer. Anal. 31 (2011) 61–85. | MR | Zbl | DOI
[8] and , Large-time behaviour of a family of finite volume schemes for boundary-driven convection–diffusion equations. IMA J. Numer. Anal. 40 (2020) 2473–2504. | MR | DOI
[9] and , A practical difference scheme for fokker-planck equations. J. Comput. Phys. 6 (1970) 1–16. | Zbl | DOI
[10] , , and , Fokker–Planck equations for a free energy functional or Markov process on a graph. 203 (2012) 969–1008. | MR | Zbl
[11] and , A third strang lemma and an Aubin-Nitsche trick for schemes in fully discrete formulation. Calcolo 55 (2018) 40. | MR | DOI
[12] and , On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Networks Heterg. Media 10 (2015) 233–253. | MR | DOI
[13] , , and , Inferring transition rates of networks from populations in continuous-time markov processes. J. Chem. Theory Comput. 11 (2015) 5464–5472. | DOI
[14] , and , Markov models from the square root approximation of the Fokker-Planck equation: calculating the grid-dependent flux. J. Phys. Condensed Matter 33 (2021) 115902. | DOI
[15] , and , A gradient system with a wiggly energy and relaxed EDP-convergence. ESAIM Control Optim. Calc. Var. 25 (2019) 68. | MR | Numdam | DOI
[16] and , Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ratio. Mech. Anal. 206 (2012) 997–1038. | MR | Zbl | DOI
[17] , and , Finite volume methods. Handb. Numer. Anal. 7 (2000) 713–1018. | MR | Zbl
[18] , and , A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local dirichlet problems. Numer. Math. 102 (2006) 463–495. | MR | Zbl | DOI
[19] , , , , and , From metastable to coherent sets-time-discretization schemes. Chaos Interdisciplinary J. Nonlinear Sci. 29 (2019) 012101. | MR | DOI
[20] , and , Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics. J. Comput. Phys. 346 (2017) 497–513. | MR | DOI
[21] , , , , and , Drift-Diffusion Models. In: Handbook of Optoelectronic Device Modeling and Simulation: Lasers, Modulators, Photodetectors, Solar Cells, and Numerical Methods, edited by . Vol. 2, Chapter 50 CRC Press, Taylor & Francis Group, Boca Raton (2017) 731–771.
[22] , and , Evolutionary -convergence of entropic gradient flow structures for fokker-planck equations in multiple dimensions. Preprint (2020). | arXiv | MR
[23] and , Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discr. Cont. Dyn. Syst.-S. 14 (2021) 395. | MR
[24] , and , Error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions. SIAM J. Numer. Anal. 37 (2000) 1935–1972. | MR | Zbl | DOI
[25] , , and , Homogenisation of one-dimensional discrete optimal transport. J. Math. Pures App. 139 (2020) 204–234. | MR | DOI
[26] , Convergences of the squareroot approximation scheme to the Fokker-Planck operator. Math. Models Methods Appl. Sci. 28 (2018) 2599–2635. | MR | DOI
[27] , Stochastic homogenization on randomly perforated domains. Preprint (2020). | arXiv
[28] , and , On the development and generalizations of Allen-Cahn and Stefan equations within a thermodynmic framework. Z. Angew. Math. Phys. (ZAMP) 63 (2012) 759–776. | MR | Zbl | DOI
[29] , Differencing scheme for a differential equation with a small parameter affecting the highest derivative. Math. Notes Acad. Sci. USSR 6 (1969) 237–248. Translated from Mat. Zametki 6 (1969) 237–248. | MR | Zbl
[30] , and , The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1–17. | MR | Zbl | DOI
[31] , Generalized Scharfetter-Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient. J. Comput. Phys. 402 (2020) 109091. | MR | DOI
[32] , and , Finite volume methods for convection-diffusion problems. SIAM J. Numer. Anal. 33 (1996) 31–55. | MR | Zbl | DOI
[33] , and , A square root approximation of transition rates for a markov state model. SIAM J. Matrix Anal. App. 34 (2013) 738–756. | MR | Zbl | DOI
[34] , , and , On microscopic origins of generalized gradient structures. Discr. Cont. Dynam. Syst. Ser. S 10 (2017) 1–35. | MR
[35] and , Large time behaviors of upwind schemes and -schemes for Fokker-Planck equations on by jump processes. Math. Comput. 89 (2020) 2283–2320. | MR | DOI
[36] , Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261 (2011) 2250–2292. | MR | Zbl | DOI
[37] , Contribution a l’étude de la cinétique physico-chimique. Ann. Phys. III (1915) 120–231. | DOI
[38] , The Stationary Semiconductor Device Equations. Springer, Vienna (1986). | MR | DOI
[39] , A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24 (2011) 1329–1346. | MR | Zbl | DOI
[40] , Geodesic convexity of the relative entropy in reversible markov chains. Calc. Var. Part. Differ. Equ. 48 (2013) 1–31. | MR | Zbl | DOI
[41] , On evolutionary -convergence for gradient systems (Ch. 3). In: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, edited by , and . Proc. of Summer School in Twente University, June 2012. Vol. 3 of Lecture Notes in Applied Math. Mechanics Springer (2016) 187–249. | MR
[42] and , Coarse-graining via EDP-convergence for linear fast-slow reaction systems. Math. Models Methods Appl. Sci. 30 (2020) 1765–1807. | MR | DOI
[43] , and , On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion. Potential Anal. 41 (2014) 1293–1327. | MR | Zbl | DOI
[44] , , and , Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics. SIAM J. Appl. Math. 77 (2017) 1562–1585. | MR | DOI
[45] and , An analysis of the Scharfetter-Gummel box method for the stationary semiconductor device equations. ESAIM: M2AN 28 (1994) 123–140. | MR | Zbl | Numdam | DOI
[46] , , and , Jump processes as generalized gradient flows. Preprint (2020). | arXiv | MR
[47] and , Large-signal analysis of a silicon read diode oscillator. IEEE Trans. Electron. Devices 16 (1969) 64–77. | DOI
[48] , Generalizations of the logarithmic mean. Math. Mag. 48 (1975) 87–92. | MR | Zbl | DOI
[49] , Theory of the flow of electrons and holes in germanium and other semiconductors. Bell Syst. Tech. J. 29 (1950) 560–607. | DOI
[50] , and , Finite difference schemes on triangular cell-centered grids with local refinement. SIAM J. Sci. Stat. Comput. 13 (1992) 1287–1313. | MR | Zbl | DOI
[51] and , A fuzzy-set theoretical framework for computing exit rates of rare events in potential-driven diffusion processes. Preprint (2017). | arXiv
[52] Wolfram Mathematica, Version 11.1. Wolfram Research. Inc., Champagne, IL, USA (2017).
[53] and , A monotone finite element scheme for converction-diffusion equations. Math. Comput. 68 (1999) 1429–1446. | MR | Zbl | DOI
Cité par Sources :
