Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balance. (English) Zbl 07516278
Summary: We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu, C. Wang, and Y. Wang, J. Comput. Phys., 436 (2021), 110253] for a reaction-diffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. The scheme is energy stable and positivity-preserving. In this paper, the detailed convergence analysis and error estimate are performed for the operator splitting scheme. The nonlinearity in the reaction trajectory equation, as well as the implicit treatment of nonlinear and singular logarithmic terms, impose challenges in numerical analysis. To overcome these difficulties, we make use of the convex nature of the logarithmic nonlinear terms. In addition, a combination of rough error estimate and refined error estimate leads to a desired bound of the numerical error in the reaction stage, in the discrete maximum norm. Furthermore, a discrete maximum principle yields the evolution bound of the numerical error function at the diffusion stage. As a direct consequence, a combination of the numerical error analysis at different stages and the consistency estimate for the operator splitting procedure results in the convergence estimate of the numerical scheme for the full reaction-diffusion system. The convergence analysis technique could be extended to a more general class of dissipative reaction mechanisms. As an example, we also consider a near-equilibrium reaction kinetics, which was derived by the linear response assumption on the reaction trajectory. Although the reaction rate is more complicated in terms of concentration variables, we show that the numerical approach and the convergence analysis also work in this case.
MSC:
| 65-XX | Numerical analysis |
| 35K35 | Initial-boundary value problems for higher-order parabolic equations |
| 35K55 | Nonlinear parabolic equations |
| 49J40 | Variational inequalities |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
reaction-diffusion system; energetic variational formulation; operator splitting scheme; positivity preserving; optimal rate convergence analysis; rough error estimate and refined error estimateReferences:
| [1] | H. Abels and M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67 (2007), pp. 3176-3193. · Zbl 1121.35018 |
| [2] | D. F. Anderson, G. Craciun, M. Gopalkrishnan, and C. Wiuf, Lyapunov functions, stationary distributions, and non-equilibrium potential for reaction networks, Bull. Math. Biol., 77 (2015), pp. 1744-1767. · Zbl 1339.92102 |
| [3] | D. N. Arnold, Stability, consistency, and convergence of numerical discretizations, in Encyclopedia of Applied and Computational Mathematics, B. Engquist, ed., Springer, Berlin, Heidelberg, pp. 1358-1364, https://doi.org/10.1007/978-3-540-70529-1_407. |
| [4] | S. Badia, R. Planas, and J. V. Gutiérrez-Santacreu, Unconditionally stable operator splitting algorithms for the incompressible magnetohydrodynamics system discretized by a stabilized finite element formulation based on projections, Internat. J. Numer. Methods Engrg., 93 (2013), pp. 302-328. · Zbl 1352.76122 |
| [5] | W. Bao, S. Jin, and P. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), pp. 487-524. · Zbl 1006.65112 |
| [6] | A. Bátkai, P. Csomós, and B. Farkas, Operator splitting for nonautonomous delay equations, Comput. Math. Appl., 65 (2013), pp. 315-324. · Zbl 1319.65054 |
| [7] | C. Besse, B. Bidégaray, and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), pp. 26-40, https://doi.org/10.1137/S0036142900381497. · Zbl 1026.65073 |
| [8] | W. Chen, C. Wang, X. Wang, and S. Wise, Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential, J. Comput. Phys. X, 3 (2019), 100031. · Zbl 07785514 |
| [9] | M. Chipot, D. Kinderlehrer, and M. Kowalczyk, A variational principle for molecular motors, Meccanica, 38 (2003), pp. 505-518. · Zbl 1032.92005 |
| [10] | T. De Donder, L’affinité, Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8, 9 (1927), pp. 1-94. |
| [11] | A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), pp. 1491-1514. · Zbl 0831.35088 |
| [12] | S. Descombes, Convergence of a splitting method of high order for reaction-diffusion systems, Math. Comp., 70 (2001), pp. 1481-1501. · Zbl 0981.65107 |
| [13] | S. Descombes and M. Massot, Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: Singular perturbation and order reduction, Numer. Math., 97 (2004), pp. 667-698. · Zbl 1060.65105 |
| [14] | L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), pp. 157-176. · Zbl 1096.35018 |
| [15] | L. Desvillettes, K. Fellner, and B. Q. Tang, Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks, SIAM J. Math. Anal., 49 (2017), pp. 2666-2709, https://doi.org/10.1137/16M1073935. · Zbl 1368.35036 |
| [16] | L. Dong, C. Wang, S. Wise, and Z. Zhang, A positivity-preserving, energy stable scheme for a ternary Cahn-Hilliard system with the singular interfacial parameters, J. Comput. Phys., 442 (2021), 110451. · Zbl 07513797 |
| [17] | L. Dong, C. Wang, H. Zhang, and Z. Zhang, A positivity-preserving, energy stable and convergent numerical scheme for the Cahn-Hilliard equation with a Flory-Huggins-deGennes energy, Commun. Math. Sci., 17 (2019), pp. 921-939. · Zbl 1423.60052 |
| [18] | L. Dong, C. Wang, H. Zhang, and Z. Zhang, A positivity-preserving second-order BDF scheme for the Cahn-Hilliard equation with variable interfacial parameters, Commun. Comput. Phys., 28 (2020), pp. 967-998. · Zbl 1528.65045 |
| [19] | C. Duan, C. Liu, C. Wang, and X. Yue, Convergence analysis of a numerical scheme for the porous medium equation by an energetic variational approach, Numer. Math. Theoret. Methods Appl., 13 (2020), pp. 63-80. · Zbl 1463.65333 |
| [20] | L. Einkemmer and A. Ostermann, An almost symmetric Strang splitting scheme for nonlinear evolution equations, Comput. Math. Appl., 67 (2014), pp. 2144-2157. · Zbl 1368.65074 |
| [21] | L. Einkemmer and A. Ostermann, Convergence analysis of Strang splitting for Vlasov-type equations, SIAM J. Numer. Anal., 52 (2014), pp. 140-155, https://doi.org/10.1137/130918599. · Zbl 1297.65106 |
| [22] | C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), pp. 404-423, https://doi.org/10.1137/S0036141094267662. · Zbl 0856.35071 |
| [23] | M.-H. Giga, A. Kirshtein, and C. Liu, Variational Modeling and Complex Fluids, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, 2018, pp. 73-113. |
| [24] | A. Giorgini, M. Grasselli, and A. Miranville, The Cahn-Hiliard-Ono equation with singular potential, Math. Models Methods Appl. Sci., 27 (2017), pp. 2485-2510. · Zbl 1386.35023 |
| [25] | A. Glitzky and A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. Angew. Math. Phys., 64 (2013), pp. 29-52. · Zbl 1270.35256 |
| [26] | W. Hao and C. Xue, Spatial pattern formation in reaction-diffusion models: A computational approach, J. Math. Biol., 80 (2020), pp. 521-543. · Zbl 1432.92009 |
| [27] | J. Haskovec, S. Hittmeir, P. Markowich, and A. Mielke, Decay to equilibrium for energy-reaction-diffusion systems, SIAM J. Math. Anal., 50 (2018), pp. 1037-1075, https://doi.org/10.1137/16M1062065. · Zbl 1387.35352 |
| [28] | A. Hawkins-Daarud, K. G. van der Zee, and J. Tinsley Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012), pp. 3-24. · Zbl 1242.92030 |
| [29] | E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Courier Corporation, 2012. |
| [30] | F. Jülicher, A. Ajdari, and J. Prost, Modeling molecular motors, Rev. Modern Phys., 69 (1997), pp. 1269-1282. |
| [31] | M. Koleva and L. Vulkov, Operator splitting kernel based numerical method for a generalized Leland’s model, J. Comput. Appl. Math., 275 (2015), pp. 294-303. · Zbl 1334.65132 |
| [32] | S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), pp. 1616-1620. · Zbl 1226.35077 |
| [33] | H. G. Lee and J. Y. Lee, A second order operator splitting method for Allen-Cahn type equations with nonlinear source terms, Phys. A, 432 (2015), pp. 24-34. · Zbl 1400.82256 |
| [34] | X. Li, Z. Qiao, and H. Zhang, Convergence of a fast explicit operator splitting method for the epitaxial growth model with slope selection, SIAM J. Numer. Anal., 55 (2017), pp. 265-285, https://doi.org/10.1137/15M1041122. · Zbl 1368.35268 |
| [35] | M. Liero and A. Mielke, Gradient structures and geodesic convexity for reaction-diffusion systems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371 (2013), 20120346. · Zbl 1292.35149 |
| [36] | C. Liu and J.-E. Sulzbach, Well-posedness for the Reaction-diffusion Equation with Temperature in a Critical Besov Space, preprint, https://arxiv.org/abs/2101.10419, 2021. |
| [37] | C. Liu, C. Wang, and Y. Wang, A structure-preserving, operator splitting scheme for reaction-diffusion equations involving the law of mass action, J. Comput. Phys., 436 (2021), 110253. · Zbl 07513840 |
| [38] | C. Liu, C. Wang, S. M. Wise, X. Yue, and S. Zhou, A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system, Math. Comp., 90 (2021), pp. 2071-2106. · Zbl 1480.65213 |
| [39] | 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. Comput. Phys., 364 (2018), pp. 73-94. · Zbl 1395.65050 |
| [40] | C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), pp. 2141-2153. · Zbl 1198.65186 |
| [41] | A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), pp. 1329-1346. · Zbl 1227.35161 |
| [42] | A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), pp. 479-499. · Zbl 1262.35127 |
| [43] | A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), pp. 545-582. · Zbl 1050.35113 |
| [44] | L. Onsager, Reciprocal relations in irreversible processes. I, Phys. Rev., 37 (1931), pp. 405-426. · JFM 57.1168.10 |
| [45] | L. Onsager, Reciprocal relations in irreversible processes. II, Phys. Rev., 38 (1931), pp. 2265-2279. · Zbl 0004.18303 |
| [46] | J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), pp. 189-192. |
| [47] | B. Perthame, F. Quirós, and J. L. Vázquez, The Hele-Shaw asymptotics for mechanical models of tumor growth, Arch. Ration. Mech. Anal., 212 (2014), pp. 93-127. · Zbl 1293.35347 |
| [48] | J. Prost, F. Jülicher, and J.-F. Joanny, Active gel physics, Nature Phys., 11 (2015), pp. 111-117. |
| [49] | Y. Qian, C. Wang, and S. Zhou, A positive and energy stable numerical scheme for the Poisson-Nernst-Planck-Cahn-Hilliard equations with steric interactions, J. Comput. Phys., 426 (2021), p. 109908. · Zbl 07510042 |
| [50] | D. Shear, An analog of the Boltzmann \(h\)-theorem (a Liapunov function) for systems of coupled chemical reactions, J. Theoret. Biol., 16 (1967), pp. 212-228. |
| [51] | J. Shen and Z. Wang, Error analysis of the Strang time-splitting Laguerre-Hermite/Hermite collocation methods for the Gross-Pitaevskii equation, Found. Comput. Math., 13 (2013), pp. 99-137. · Zbl 1264.65168 |
| [52] | M. Thalhammer, Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations, SIAM J. Numer. Anal., 50 (2012), pp. 3231-3258, https://doi.org/10.1137/120866373. · Zbl 1267.65116 |
| [53] | H. Wang, C. S. Peskin, and T. C. Elston, A robust numerical algorithm for studying biomolecular transport processes, J. Theoret. Biol., 221 (2003), pp. 491-511. · Zbl 1464.92119 |
| [54] | Y. Wang, C. Liu, P. Liu, and B. Eisenberg, Field theory of reaction-diffusion: Mass action with an energetic variational approach, Phys. Rev. E, 102 (2020), 062147. |
| [55] | Y. Wang, T.-F. Zhang, and C. Liu, A two species micro-macro model of wormlike micellar solutions and its maximum entropy closure approximations: An energetic variational approach, J. Non-Newtonian Fluid Mech., 293 (2021), 104559. |
| [56] | J. Wei, Axiomatic treatment of chemical reaction systems, J. Chem. Phys., 36 (1962), pp. 1578-1584. |
| [57] | M. Yuan, W. Chen, C. Wang, S. Wise, and Z. Zhang, An energy stable finite element scheme for the three-component Cahn-Hilliard-type model for macromolecular microsphere composite hydrogels, J. Sci. Comput., 87 (2021), 78. · Zbl 1473.65219 |
| [58] | C. Zhang, J. Huang, C. Wang, and X. Yue, On the operator splitting and integral equation preconditioned deferred correction methods for the “good” Boussinesq equation, J. Sci. Comput., 75 (2018), pp. 687-712. · Zbl 1398.65167 |
| [59] | C. Zhang, H. Wang, J. Huang, C. Wang, and X. Yue, A second order operator splitting numerical scheme for the “good” Boussinesq equation, Appl. Numer. Math., 119 (2017), pp. 179-193. · Zbl 1368.65203 |
| [60] | J. Zhang, C. Wang, S. M. Wise, and Z. Zhang, Structure-preserving, energy stable numerical schemes for a liquid thin film coarsening model, SIAM J. Sci. Comput., 43 (2021), pp. A1248-A1272, https://doi.org/10.1137/20M1375656. · Zbl 1468.65119 |
| [61] | S. Zhao, Operator splitting ADI schemes for pseudo-time coupled nonlinear solvation simulations, J. Comput. Phys., 257 (2014), pp. 1000-1021. · Zbl 1349.78105 |
| [62] | S. Zhao, J. Ovadia, X. Liu, Y.-T. Zhang, and Q. Nie, Operator splitting implicit integration factor methods for stiff reaction-diffusion-advection systems, J. Comput. Phys., 230 (2011), pp. 5996-6009. · Zbl 1220.65120 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
