We study a finite volume scheme for the approximation of the solution to convection diffusion equations with nonlinear convection and Robin boundary conditions. The scheme builds on the interpretation of such a continuous equation as the hydrodynamic limit of some simple exclusion jump process. We show that the scheme admits a unique discrete solution, that the natural bounds on the solution are preserved, and that it encodes the second principle of thermodynamics in the sense that some free energy is dissipated along time. The convergence of the scheme is then rigorously established thanks to compactness arguments. Numerical simulations are finally provided, highlighting the overall good behavior of the scheme.
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Clément Cancès 1; Juliette Venel 2
@article{CRMATH_2023__361_G2_535_0, author = {Cl\'ement Canc\`es and Juliette Venel}, title = {On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations}, journal = {Comptes Rendus. Math\'ematique}, pages = {535--558}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.421}, language = {en}, }
TY - JOUR AU - Clément Cancès AU - Juliette Venel TI - On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations JO - Comptes Rendus. Mathématique PY - 2023 SP - 535 EP - 558 VL - 361 PB - Académie des sciences, Paris DO - 10.5802/crmath.421 LA - en ID - CRMATH_2023__361_G2_535_0 ER -
Clément Cancès; Juliette Venel. On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 535-558. doi : 10.5802/crmath.421. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.421/
[1] Numerical analysis of a finite volume scheme for a seawater intrusion model with cross-diffusion in an unconfined aquifer, Numer. Methods Partial Differ. Equations, Volume 34 (2018) no. 3, pp. 857-880 | DOI | Zbl
[2] A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs, J. Funct. Anal., Volume 273 (2017) no. 12, pp. 3633-3670 | DOI | Zbl
[3] An augmented Lagrangian approach to Wasserstein gradient flows and applications, Gradient flows: from theory to application (ESAIM Proc. Surveys), Volume 54, EDP Sciences, Les Ulis, 2016, pp. 1-17 | Zbl
[4] Lyapunov functionals for boundary-driven nonlinear drift–diffusion equations, Nonlinearity, Volume 27 (2014) no. 9, pp. 2111-2132 | DOI | Zbl
[5] Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure, Comput. Geosci., Volume 17 (2013) no. 3, pp. 573-597 | DOI | Zbl
[6] On the Chang and Cooper scheme applied to a linear Fokker-Planck equation, Commun. Math. Sci., Volume 8 (2010) no. 4, pp. 1079-1090 | DOI | Zbl
[7] Nonlinear cross-diffusion with size-exclusion, SIAM J. Math. Anal., Volume 42 (2010) no. 6, pp. 2842-2871 | DOI | Zbl
[8] A numerical analysis focused comparison of several finite volume schemes for a unipolar degenerated drift-diffusion model, IMA J. Numer. Anal., Volume 41 (2021) no. 1, pp. 271-314 | DOI | Zbl
[9] Mathematical analysis of a thermodynamically consistent reduced model for iron corrosion (2022) (working paper or preprint, https://arxiv.org/abs/2201.13193)
[10] A variational finite volume scheme for Wasserstein gradient flows, Numer. Math., Volume 146 (2020) no. 3, pp. 437-480 | DOI | Zbl
[11] Primal dual methods for Wasserstein gradient flows, Found. Comput. Math. (2021) (Online first) | DOI | Zbl
[12] Large-time behaviour of a family of finite volume schemes for boundary-driven convection–diffusion equations, IMA J. Numer. Anal., Volume 40 (2020) no. 4, pp. 2473-2504 | DOI | Zbl
[13] Asymptotic behavior of the Scharfetter–Gummel scheme for the drift-diffusion model, Finite volumes for complex applications VI. Problems & perspectives. Volume 1, 2 (Springer Monographs in Mathematics), Volume 4, Springer, 2011, pp. 235-243 | DOI | Zbl
[14] Nonlinear functional analysis, Springer, 1985 | DOI | Zbl
[15] A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems, Numer. Math., Volume 102 (2006) no. 3, pp. 463-495 | DOI | Zbl
[16] Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes, IMA J. Numer. Anal., Volume 18 (1998) no. 4, pp. 563-594 | DOI | Zbl
[17] TP or not TP, that is the question, Comput. Geosci., Volume 18 (2014) no. 3-4, pp. 285-296 | DOI | Zbl
[18] Finite volume methods, Solution of equations in (Part 3). Techniques of scientific computing (Part 3) (P. G. Ciarlet et al., eds.) (Handbook of Numerical Analysis), Volume 7, North-Holland, 2000, pp. 713-1020 | DOI | Zbl
[19] A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure, Numer. Math., Volume 137 (2017) no. 3, pp. 535-577 | DOI | Zbl
[20] Why Do We Need Voronoi Cells and Delaunay Meshes?, Numerical Geometry, Grid Generation and Scientific Computing (Vladimir A. Garanzha; Lennard Kamenski; Hang Si, eds.) (Lecture Notes in Computational Science and Engineering), Volume 131, Springer, 2019, pp. 45-60 | DOI | Zbl
[21] Convergences of the squareroot approximation scheme to the Fokker–Planck operator, Math. Models Methods Appl. Sci., Volume 28 (2018) no. 13, pp. 2599-2635 | DOI | Zbl
[22] Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator, ESAIM, Math. Model. Numer. Anal., Volume 55 (2021) no. 6, pp. 3017-3042 | DOI | Zbl
[23] Scaling limits of interacting particle systems, Grundlehrender der Mathematischen Wissenschaften, 320, Springer, 1999 | DOI | Zbl
[24] Topologie et équations fonctionnelles, Ann. Sci. Éc. Norm. Supér., Volume 51 (1934) no. 3, pp. 45-78 | DOI | Numdam | Zbl
[25] Fisher information regularization schemes for Wasserstein gradient flows, J. Comput. Phys., Volume 416 (2020), 109449, 23 pages | Zbl
[26] A square root approximation of transition rates for a Markov state model, SIAM J. Matrix Anal. Appl., Volume 34 (2013) no. 2, pp. 738-756 | Zbl
[27] Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM, Math. Model. Numer. Anal., Volume 48 (2014) no. 3, pp. 697-726 | DOI | Numdam | Zbl
[28] A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, Volume 24 (2011) no. 4, pp. 1329-1346 | DOI | Zbl
[29] On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Anal., Volume 41 (2014) no. 4, pp. 1293-1327 | DOI | Zbl
[30] Jump processes as generalized gradient flows, Calc. Var. Partial Differ. Equ., Volume 61 (2022) no. 1, 33, 85 pages | Zbl
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