Structure Preserving Schemes for a Class of Wasserstein Gradient Flows

Expand
  • 1 Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA;
    2 School of Mathematical Science, Eastern Institute of Technology, Ningbo 315200, Zhejiang, China

Received date: 2024-03-27

  Revised date: 2024-08-28

  Accepted date: 2024-10-14

  Online published: 2025-05-23

Supported by

This work was partially supported by the NSFC (Grant No. 12371409).

Abstract

We introduce in this paper two time discretization schemes tailored for a range of Wasserstein gradient flows. These schemes are designed to preserve mass and positivity and to be uniquely solvable. In addition, they also ensure energy dissipation in many typical scenarios. Through extensive numerical experiments, we demonstrate the schemes’ robustness, accuracy, and efficiency.

Cite this article

Shiheng Zhang, Jie Shen . Structure Preserving Schemes for a Class of Wasserstein Gradient Flows[J]. Communications on Applied Mathematics and Computation, 2025 , 7(3) : 1174 -1194 . DOI: 10.1007/s42967-025-00486-2

References

1. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)
2. Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998)
3. Bailo, R., Carrillo, J.A., Hu, J.: Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient-flow structure. Commun. Math. Sci. 18, 1259–1303 (2020)
4. Bazant, M.Z., Thornton, K., Ajdari, A.: Diffuse-charge dynamics in electrochemical systems. Phys. Rev. E 70, 021506 (2004)
5. Biler, P., Hebisch, W., Nadzieja, T.: The Debye system: existence and large time behavior of solutions. Nonlinear Anal. Theory Methods Appl. 23, 1189–1209 (1994)
6. Bolley, F., Canizo, J.A., Carrillo, J.A.: Stochastic mean-field limit: non-Lipschitz forces and swarming. Math. Models Methods Appl. Sci. 21, 2179–2210 (2011)
7. Bonizzoni, F., Braukhoff, M., Jüngel, A., Perugia, I.: A structure-preserving discontinuous Galerkin scheme for the Fisher-KPP equation. Numer. Math. 146, 119–157 (2020)
8. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)
9. Carrillo, J.A., Craig, K., Wang, L., Wei, C.: Primal dual methods for Wasserstein gradient flows. Found. Comput. Math. 2022, 1–55 (2022)
10. Carrillo, J.A., Wang, L., Xu, W., Yan, M.: Variational asymptotic preserving scheme for the VlasovPoisson-Fokker-Planck system. Multiscale Model. Simul. 19, 478–505 (2021)
11. Doi, M.: Onsager’s variational principle in soft matter. J. Phys.: Condens. Matter 23, 284118 (2011)
12. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics, vol. 73. Oxford University Press, Oxford (1988)
13. Duan, C., Chen, W., Liu, C., Wang, C., Zhou, S.: Convergence analysis of structure-preserving numerical methods for nonlinear Fokker-Planck equations with nonlocal interactions. Math. Methods Appl. Sci. 45, 3764–3781 (2022)
14. Duan, C., Chen, W., Liu, C., Yue, X., Zhou, S.: Structure-preserving numerical methods for nonlinear Fokker-Planck equations with nonlocal interactions by an energetic variational approach. SIAM J. Sci. Comput. 43, B82–B107 (2021)
15. Elder, K., Katakowski, M., Haataja, M., Grant, M.: Modeling elasticity in crystal growth. Phys. Rev. Lett. 88, 245701 (2002)
16. Elliott, C.M., Stuart, A.: The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30, 1622–1663 (1993)
17. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. MRS Online Proc. Lib. (OPL) 529, 39 (1998)
18. Fu, G., Liu, S., Osher, S., Li, W.: High order computation of optimal transport, mean field planning, and mean field games. arXiv:2302.02308 (2023)
19. Fu, G., Osher, S., Li, W.: High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems. arXiv:2303.08950 (2023)
20. Gu, Y., Shen, J.: Bound preserving and energy dissipative schemes for porous medium equation. J. Comput. Phys. 410, 109378 (2020)
21. Gurtin, M.E., Polignone, D., Vinals, J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6, 815–831 (1996)
22. Harker, P., Pang, J.-S.: A damped-Newton method for the linear complementarity problem. Numer. Algorithms 26(01), 265–284 (1990)
23. Hu, J., Zhang, X.: Positivity-preserving and energy-dissipative finite difference schemes for the FokkerPlanck and Keller-Segel equations. IMA J. Numer. Anal. 43, 1450–1484 (2023)
24. Jacobs, M., Lee, W., Léger, F.: The back-and-forth method for Wasserstein gradient flows. ESAIM Control Optim. Calc. Var. 27, 28 (2021)
25. Jacobs, M., Léger, F.: A fast approach to optimal transport: the back-and-forth method. Numer. Math. 146, 513–544 (2020)
26. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998)
27. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
28. Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)
29. Leclerc, H., Mérigot, Q., Santambrogio, F., Stra, F.: Lagrangian discretization of crowd motion and linear diffusion. SIAM J. Numer. Anal. 58, 2093–2118 (2020)
30. Leslie, F.M.: Theory of flow phenomena in liquid crystals. In: Advances in Liquid Crystals, vol. 4, pp. 1–81. Elsevier, London (1979)
31. Li, W., Lee, W., Osher, S.: Computational mean-field information dynamics associated with reactiondiffusion equations. J. Comput. Phys. 466, 111409 (2022)
32. Patlak, C.S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)
33. Peletier, M.A.: Variational modelling: energies, gradient flows, and large deviations. arXiv:1402.1990 (2014)
34. Peyré, G.: Entropic approximation of Wasserstein gradient flows. SIAM J. Imaging Sci. 8, 2323–2351 (2015)
35. Shen, J., Wang, C., Wang, X., Wise, S.M.: Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy. SIAM J. Numer. Anal. 50, 105–125 (2012)
36. Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J. Numer. Anal. 56, 2895–2912 (2018)
37. Shen, J., Xu, J.: Unconditionally bound preserving and energy dissipative schemes for a class of KellerSegel equations. SIAM J. Numer. Anal. 58, 1674–1695 (2020)
38. Shen, J., Xu, J.: Unconditionally positivity preserving and energy dissipative schemes for Poisson-NernstPlanck equations. Numer. Math. 148, 671–697 (2021)
39. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)
40. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61, 474–506 (2019)
41. Sznitman, A.-S.: Topics in propagation of chaos. Lect. Notes Math. 1991, 165–251 (1991)
42. Vázquez, J.L.: An introduction to the mathematical theory of the porous medium equation. In: Shape Optimization and Free Boundaries, pp. 347–389. Springer, London (1992)
43. Vázquez, J.L.: The Porous Medium Equation: Mathematical Theory. Oxford University Press, Oxford (2007)
44. Villani, C.: Topics in Optimal Transportation, vol. 58. American Mathematical Soc., London (2021)
45. Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294–316 (2016)
46. Yue, P., Feng, J.J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317 (2004)
47. Zhao, J., Wang, Q., Yang, X.: Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. Int. J. Numer. Methods Eng. 110, 279–300 (2017)
Options
Outlines

/