Structure-preserving scheme for two-phase convection reaction diffusion system
Structure-preserving scheme for two-phase convection reaction diffusion system
DOI:
https://doi.org/10.56143/ujmcs.v1i2.8Keywords:
free boundary problem, advection, reaction, diffusion, structure-preserving method, stability, numerical simulation.Abstract
In this paper, we introduce a novel structure-preserving explicit numerical scheme for a two-phase convection reaction diffusion system featuring a dynamically evolving interface. A priori estimates in H\"older norms are established for both the
solution and the free boundary, which allow us to prove the existence and
uniqueness of a classical solution and to analyze its qualitative properties. We also present a comparative study of three numerical approaches: the upwind
implicit method, the Crank--Nicolson scheme, and the proposed explicit scheme.
Numerical experiments demonstrate the robustness and stability of the new method,
even in regimes dominated by strong advection and highly nonlinear reaction terms.
The proposed scheme provides physically reliable results and is suitable for
modeling interface-driven processes arising in applications such as
osseointegration around dental implants, biological invasion, and sharp-interface
phase transition phenomena.
References
[1] Du, Y. and Lin, Z.G.: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal., 42, 377–405 (2010).
[2] Elmurodov, A. N. and Sotvoldiyev, A. I. : A diffusive Leslie–Gower type Predator-Prey Model with two different free boundaries. Lobachevski J.Math. 44, 4254–4270 (2023).
[3] Friedman, A. : Free boundary problems in biology. Philos. Trans. A. Math. Phys. Eng. Sci. Sep 13; 373(2050):20140368. (2015). https://doi:10.1098/rsta.2014.0368.
[4] Rasulov, M. S. and Elmurodov, A. N.: A free boundary problem for a Predator-Prey System. Lobachevskii J.Math. 44, 2898–2909 (2023).
[5] Takhirov, J. O.: A free boundary problem for a reaction-diffusion equation appearing in biology. Indian J. Pure Appl. Math., 50(1): 95-112, March (2019).
[6] Elmurodov, A. N. : Two-phase problem with a free boundary for systems of parabolic equations with a nonlinear term of convection. Vestnik KRAUNC. Fiziko-Matematicheskie Nauki, 36(3), 110–122 (2021).
[7] Takhirov, J. O. and Elmurodov, A. N. : About mathematical model with a free boundary of water basins pollutions. Uzbek Math. J. 64 (4), 149–160 (2020). https://DOI:10.29229/uzmj.2020-4-16
[8] Tyson, J. J., Chen, C. K. and Novak, B.: Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr. Opin. Cell Biol.15 (2), 221-231 (2003). https://doi.org/10.1016/S0955-0674(03)00017-6
[9] Byrne, H. M., Preziosi, L. : Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol.20(4):341-66 (2003). https://DOI: 10.1093/imammb/20.4.341
[10] Wu, Z. and Zhao, J.: Two-phase free boundary problems for reaction-diffusion systems. J. Differ. Equ., 272, 1–35 (2021).
[11] Shahid, N., Ahmedet, N. and Baleanu, D.: Novel numerical analysis for nonlinear advection–reaction–diffusion systems. Comp. Math. Appl. 18(1):112-125 (2020). https://DOI:10.1515/phys-2020-0011
[12] Friedman, A., Reitich, F. : Analysis of a mathematical model for the growth of tumors. J. Math. Biol. 38(3):262–84. (1999). https://doi:10.1007/s002850050149
[13] Caffarelli, L. A., Friedman, A. : Continuity of the temperature in the Stefan problem. Indiana University Mathematics Journal, 28(1), 53–70 (1979).
[14] Mickens, R. E. : Advances in the Applications of Nonstandard Finite Difference Schemes, 2005 World Scientific, Singapore, (2005). https://DOI:10.1142/5884
[15] Anguelov, R., Lubuma, J. M.-S. : Contributions to the mathematics of the nonstandard finite difference method. Numer. Methods Partial Differential Equations, 17 (5), pp. 518-543 (2001). https://DOI:10.1002/num.1025
[16] Chen-Charpentier, B. M., Kojouharov, H. V. : Unconditionally positive finite difference scheme for advection–reaction–diffusion equations. J. Comput. Appl. Math.(2016).
[17] Friedman, A. Variational Principles and Free-Boundary Problems, Wiley-Interscience, New York (1982).
[18] Cannon, J. R. : The One-Dimensional Heat Equation. Addison-Wesley. (1984).
[19] Elmurodov, A.N.: The two-phase Stefan problem for parabolic equations. Uzbek Mathematical Journal, 4, pp.54-64 (2019). https://DOI:10.29229/uzmj.2019-4-6
[20] Elmurodov, A.N. and Rasulov, M.S.: On a Uniqueness of Solution for a Reaction-Diffusion Type System with a Free Boundary. Lobachevskii Journal of Mathematics, 43, 2099–2106 (2022).
[21] Kruzhkov, S.N.: Nonlinear parabolic equations in two independent variables. Trans. Moscow Math. Soc., 16, 355–373 (1967).
[22] Ladyzenskaja, O.A., Solonnikov, V.A. and Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type, Nauka, Moscow (1968).
[23] Chen, G., Charpentier, M., Kojouharov, H. V.: A positivity-preserving finite difference scheme for a class of nonlinear parabolic equations. Journal of Computational and Applied Mathematics, 346, 137–148 (2019).
