Open Access
Issue
ESAIM: M2AN
Volume 56, Number 1, January-February 2022
Page(s) 121 - 150
DOI https://doi.org/10.1051/m2an/2021080
Published online 07 February 2022
  1. L. Almeida, F. Bubba, B. Perthame and C. Pouchol, Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations. Netw. Heterogen. Media 14 (2019) 23–41. [CrossRef] [Google Scholar]
  2. D.G. Aronson, The focusing problem for the porous medium equation: experiment, simulation and analysis. Nonlinear Partial Differential Equations, in honor of Juan Luis Vázquez for his 70th birthday. Nonlinear Anal. 137 (2016) 135–147. [CrossRef] [MathSciNet] [Google Scholar]
  3. D.G. Aronson and J.L. Graveleau, A selfsimilar solution to the focusing problem for the porous medium equation. Eur. J. Appl. Math. 4 (1993) 65–81. [CrossRef] [Google Scholar]
  4. D.G. Aronson, O. Gil and J.L. Vázquez, Limit behaviour of focusing solutions to nonlinear diffusions. Comm. Part. Differ. Equ. 23 (1998) 307–332. [Google Scholar]
  5. M.J. Baines, M.E. Hubbard and P.K. Jimack, A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries. Appl. Numer. Math. 54 (2005) 450–469. [CrossRef] [MathSciNet] [Google Scholar]
  6. M.J. Baines, M.E. Hubbard, P.K. Jimack and A.C. Jones, Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions. Appl. Numer. Math. 56 (2006) 230–252. [CrossRef] [MathSciNet] [Google Scholar]
  7. E.D. Benedetto and D. Hoff, An interface tracking algorithm for the porous medium equation. Trans. Am. Math. Soc. 284 (1984) 463–500. [CrossRef] [Google Scholar]
  8. M. Bessemoulin-Chatard and F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34 (2012) B559–B583. [Google Scholar]
  9. F. Bubba, B. Perthame, C. Pouchol and M. Schmidtchen, Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues. Arch. Ration. Mech. Anal. 236 (2020) 735–766. [CrossRef] [MathSciNet] [Google Scholar]
  10. C.J. Budd, G.J. Collins, W.Z. Huang and R.D. Russell, Self-similar numerical solutions of the porous-medium equation using moving mesh methods. Philos. T. Roy. Soc. A 357 (1999) 1047–1077. [CrossRef] [Google Scholar]
  11. J.A. Carrillo, H. Ranetbauer and M.-T. Wolfram, Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms. J. Comp. Phys. 327 (2016) 186–202. [CrossRef] [Google Scholar]
  12. J.A. Carrillo, Y. Huang, F.S. Patacchini and G. Wolansky, Numerical study of a particle method for gradient flows. Kinet. Relat. Mod. 10 (2017) 613–641. [CrossRef] [Google Scholar]
  13. J.A. Carrillo, B. Düring, D. Matthes and D. McCormick, A lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes. J. Sci. Comp. 75 (2018) 463–1499. [Google Scholar]
  14. F. Cavalli, G. Naldi, G. Puppo and M. Semplice, High-order relaxation schemes for nonlinear degenerate diffusion problems. SIAM J. Numer. Analy. 45 (2007) 2098–2119. [CrossRef] [Google Scholar]
  15. N. David and B. Perthame, Free boundary limit of a tumor growth model with nutrient. J. Math. App. 155 (2021) 62–82. [Google Scholar]
  16. N. David and M. Schmidtchen, On the incompressible limit for a tumour growth model incorporating convective effects. Preprint arXiv:2103.02564 (2021). [Google Scholar]
  17. N. David, T. Debiec and B. Perthame, Convergence rate for the incompressible limit of nonlinear diffusion-advection equations. Preprint arXiv:2108.00787 (2021). [Google Scholar]
  18. T. Dębiec, B. Perthame, M. Schmidtchen and N. Vauchelet, Incompressible limit for a two-species model with coupling through Brinkman’s law in any dimension. J. Math. App. 145 (2021) 204–239. [Google Scholar]
  19. P. Degond, S. Hecht and N. Vauchelet, Incompressible limit of a continuum model of tissue growth for two cell populations. Netw. Heterogen. Media 15 (2020) 57–85. [CrossRef] [Google Scholar]
  20. R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41–82. [CrossRef] [MathSciNet] [Google Scholar]
  21. J.L. Graveleau and P. Jamet, A finite difference approach to some degenerate nonlinear parabolic equation. SIAM J. Appl. Math. 20 (1971) 199–223. [CrossRef] [MathSciNet] [Google Scholar]
  22. S. Jin and Z.P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commu. Pure Appl. Math. 48 (1995) 235–277. [CrossRef] [Google Scholar]
  23. I. Kim and N. Požár, Porous medium equation to Hele-Shaw flow with general initial density. Trans. Amer. Math. Soc. 370 (2018) 873–909. [Google Scholar]
  24. C. Liu and Y. Wang, On lagrangian schemes for porous medium type generalized diffusion equations: a discrete energetic variational approach. J. Comp. Phys. 417 (2020) 109566. [CrossRef] [Google Scholar]
  25. Y. Liu, C.-W. Shu and M. Zhang, High order finite difference weno schemes for nonlinear degenerate parabolic equations. SIAM J. Sci. Comp. 33 (2011) 939–965. [CrossRef] [Google Scholar]
  26. J.-G. Liu, M. Tang, L. Wang and Z. Zhou, An accurate front capturing scheme for tumor growth models with a free boundary limit. J. Comp. Phys. 364 (2018) 73–94. [CrossRef] [Google Scholar]
  27. J.-G. Liu, M. Tang, L. Wang and Z. Zhou, Analysis and computation of some tumor growth models with nutrient: from cell density models to free boundary dynamics. Discrete Continuous Dyn. Syst. B 24 (2019) 3011. [CrossRef] [MathSciNet] [Google Scholar]
  28. P. Macklin, S. McDougall, A.R.A. Anderson, M.A.J. Chaplain, V. Cristini and J. Lowengrub, Multiscale modelling and nonlinear simulation of vascular tumour growth. J. Math. Biol. 58 (2009) 765–798. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  29. L. Monsaingeon, An explicit finite-difference scheme for one-dimensional generalized porous medium equations: interface tracking and the hole filling problem. ESAIM: M2AN 50 (2016) 1011–1033. [CrossRef] [EDP Sciences] [Google Scholar]
  30. G. Naldi, L. Pareschi and G. Toscani, Relaxation schemes for partial differential equations and applications to degenerate diffusion problems. Surv. Math. Ind. 10 (2002) 315–343. [Google Scholar]
  31. C. Ngo and W. Huang, A study on moving mesh finite element solution of the porous medium equation. J. Comp. Phys. 331 (2017) 357–380. [CrossRef] [Google Scholar]
  32. B. Perthame and N. Vauchelet, Incompressible limit of a mechanical model of tumour growth with viscosity. Philos. Trans. Roy. Soc. A 373 (2015) 20140283. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  33. B. Perthame, F. Quirós and J.L. Vázquez, The Hele-Shaw asymptotics for mechanical models of tumor growth. Arch. Ration. Mech. Anal. 212 (2014) 93–127. [Google Scholar]
  34. B. Perthame, F. Quirós, M. Tang and N. Vauchelet, Derivation of a Hele-Shaw type system from a cell model with active motion. Interfaces Free Bound. 16 (2014) 489–508. [CrossRef] [MathSciNet] [Google Scholar]
  35. B. Perthame, M. Tang and N. Vauchelet, Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient. Math. Models Methods Appl. Sci. 24 (2014) 2601–2626. [Google Scholar]
  36. M.E. Rose, Numerical methods for flows through porous media. I. Math. Comp. 40 (1983) 435–467. [CrossRef] [MathSciNet] [Google Scholar]
  37. Q. Zhang and Z.-L. Wu, Numerical simulation for porous medium equation by local discontinuous galerkin finite element method. J. Sci. Comp. 38 (2009) 127–148. [CrossRef] [MathSciNet] [Google Scholar]

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