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May  2020, 13(5): 1567-1587. doi: 10.3934/dcdss.2020089

## Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation

 University "Al. I. Cuza" of Iaşi, 700506, Iaşi, Romania

Received  January 2018 Revised  August 2018 Published  June 2019

We present the error analysis of two time-stepping schemes of fractional steps type, used in the discretization of a nonlinear reaction-diffusion equation with Neumann boundary conditions, relevant in phase transition and interface problems. We start by investigating the solvability of a such boundary value problems in the class $W^{1,2}_p(Q)$. One proves the existence, the regularity and the uniqueness of solutions, in the presence of the cubic nonlinearity type. The convergence and error estimate results (using energy methods) for the iterative schemes of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual algorithm is formulated in the end. Numerical experiments are presented in order to validates the theoretical results (conditions of numerical stability) and to compare the accuracy of the methods.

Citation: Costică Moroşanu. Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1567-1587. doi: 10.3934/dcdss.2020089
##### References:
 [1] H.-D. Alber and P. Zhu, Comparison of a rapidely converging phase field model for interfaces in solids with the Allen-Cahn model, J. Elast., 111 (2013), 153-221.  doi: 10.1007/s10659-012-9398-x.  Google Scholar [2] S. M. Allen and J. W. Cahn, A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar [3] V. Arnăutu and C. Moroşanu, Numerical approximation for the phase-field transition system, Intern. J. Com. Math., 62 (1996), 209-221.  doi: 10.1080/00207169608804538.  Google Scholar [4] T. Benincasa and C. Moroşanu, Fractional steps scheme to approximate the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions, Numer. Funct. Anal. and Optimiz., 30 (2009), 199-213.  doi: 10.1080/01630560902841120.  Google Scholar [5] T. Benincasa, A. Favini and C. Moroşanu, A Product Formula Approach to a Non-homogeneous Boundary Optimal Control Problem Governed by Nonlinear Phase-field Transition System. PART I: A Phase-field Model, J. Optim. Theory and Appl., 148 (2011), 14-30.  doi: 10.1007/s10957-010-9742-x.  Google Scholar [6] G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, Euro. Jnl of Applied Mathematics, 9 (1998), 417-445.  doi: 10.1017/S0956792598003520.  Google Scholar [7] L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 79 (2013), 12–27, arXiv: 1206.6738. doi: 10.1016/j.na.2012.11.010.  Google Scholar [8] O. Cârjă, A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions, Nonlinear Anal. TMA, 113 (2015), 190-208.  doi: 10.1016/j.na.2014.10.003.  Google Scholar [9] X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differential Equations, 206 (2004), 399-437.  doi: 10.1016/j.jde.2004.05.017.  Google Scholar [10] L. Cherfils, S. Gatti and A. Miranville, Long time behavior to the Caginalp system with singular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 11 (2012), 2261-2290.  doi: 10.3934/cpaa.2012.11.2261.  Google Scholar [11] K. Chrysafinos, Discontinuous Time-Stepping Schemes for the Allen-Cahn Equations and Applications to Optimal Control, Paper presented at the Conference on 'Advances in scientific computing and applied mathematics', Las Vegas, Nevada, October 2015. Google Scholar [12] X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math., 94 (2003), 33-65.  doi: 10.1007/s00211-002-0413-1.  Google Scholar [13] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Clarendon, Oxford, 1995. Google Scholar [14] C. Gal, M. Grasselli and A. Miranville, Non-isothermal Allen-Cahn equations with coupled dynamic boundary conditions, Nonlinear Phenomena with Energy Dissipation, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 29 (2008), 117–139.  Google Scholar [15] Gh. Iorga, C. Moroşanu and S. C. Cocindău, Numerical simulation of the solid region via phase field transition system, Metalurgia International, 8 (2008), 91-95.   Google Scholar [16] Gh. Iorga, C. Moroşanu and I. Tofan, Numerical simulation of the thickness accretions in the secondary cooling zone of a continuous casting machine, Metalurgia International, 14 (2009), 72-75.   Google Scholar [17] H. Israel, Long time behavior of an Allen-Cahn type equation with singular potential and dynamic boundary conditions, Journal of Applied Analysis and Computation, 2 (2012), 29-56.   Google Scholar [18] N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal. TMA, 22 (1994), 1163-1180.  doi: 10.1016/0362-546X(94)90235-6.  Google Scholar [19] O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasi-Linear Equations of Parabolic Type, Prov. Amer. Math. Soc., 1968. Google Scholar [20] J. L. Lions, Control of Distributed Singular Systems, Gauthier-Villars, Paris, 1985. Google Scholar [21] A. Miranville, Existence of solutions for a one-dimensional Allen-Cahn equation, J. Appl. Anal. Comput., 3 (2013), 265-277.   Google Scholar [22] A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions, Appl. Math. Modell., 40 (2016), 192-207.  doi: 10.1016/j.apm.2015.04.039.  Google Scholar [23] A. Miranville and C. Moroşanu, Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions, Discrete and Continuous Dynamical Systems Series S, 9 (2016), 537-556.  doi: 10.3934/dcdss.2016011.  Google Scholar [24] C. Moroşanu, Approximation and numerical results for phase field system by a fractional step scheme, Rev. Anal. Numér. Théor. Approx., 25 (1996), 137-151.   Google Scholar [25] C. Moroşanu, Approximation of the phase-field transition system via fractional steps method, Numer. Funct. Anal. and Optimiz., 18 (1997), 623-648.  doi: 10.1080/01630569708816782.  Google Scholar [26] C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods, Bentham Science Publishers, 2012. doi: 10.2174/97816080535061120101.  Google Scholar [27] C. Moroşanu, Qualitative and quantitative analysis for a nonlinear reaction-diffusion equation, ROMAI J., 12 (2016), 85–113. https://rj.romai.ro/arhiva/2016/2/Morosanu.pdf  Google Scholar [28] C. Moroşanu and A. Croitoru, Analysis of an iterative scheme of fractional steps type associated to the phase-field equation endowed with a general nonlinearity and Cauchy-Neumann boundary conditions, J. Math. Anal. Appl., 425 (2015), 1225-1239.  doi: 10.1016/j.jmaa.2015.01.033.  Google Scholar [29] C. Moroşanu and A.-M. Moşneagu, On the numerical approximation of the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions. Case 1D, ROMAI J., 9 (2013), 91–110, https://rj.romai.ro/arhiva/2013/1/Morosanu,Mosneagu.pdf  Google Scholar [30] C. Moroşanu and D. Motreanu, A generalized phase field system, J. Math. Anal. Appl., 237 (1999), 515-540.  doi: 10.1006/jmaa.1999.6467.  Google Scholar [31] C. Moroşanu and D. Motreanu, The phase field system with a general nonlinearity, International Journal of Differential Equations and Applications, 1 (2000), 187-204.   Google Scholar [32] C. Moroşanu, S. Pavăl and C. Trenchea, Analysis of stability and errors of three methods associated to the nonlinear reaction-diffusion equation supplied with homogeneous Neumann boundary conditions, Journal of Applied Analysis and Computation, 7 (2017), 1-19.  doi: 10.11948/2017001.  Google Scholar [33] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar [34] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for kinetics of phase transitions, Phys. D., 43 (1990), 44-62.  doi: 10.1016/0167-2789(90)90015-H.  Google Scholar [35] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences, Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [36] X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 1057-1070.  doi: 10.3934/dcdsb.2009.11.1057.  Google Scholar [37] J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM J. Sci. Comput., 31 (2009), 3042-3063.  doi: 10.1137/080738398.  Google Scholar

show all references

##### References:
 [1] H.-D. Alber and P. Zhu, Comparison of a rapidely converging phase field model for interfaces in solids with the Allen-Cahn model, J. Elast., 111 (2013), 153-221.  doi: 10.1007/s10659-012-9398-x.  Google Scholar [2] S. M. Allen and J. W. Cahn, A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar [3] V. Arnăutu and C. Moroşanu, Numerical approximation for the phase-field transition system, Intern. J. Com. Math., 62 (1996), 209-221.  doi: 10.1080/00207169608804538.  Google Scholar [4] T. Benincasa and C. Moroşanu, Fractional steps scheme to approximate the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions, Numer. Funct. Anal. and Optimiz., 30 (2009), 199-213.  doi: 10.1080/01630560902841120.  Google Scholar [5] T. Benincasa, A. Favini and C. Moroşanu, A Product Formula Approach to a Non-homogeneous Boundary Optimal Control Problem Governed by Nonlinear Phase-field Transition System. PART I: A Phase-field Model, J. Optim. Theory and Appl., 148 (2011), 14-30.  doi: 10.1007/s10957-010-9742-x.  Google Scholar [6] G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, Euro. Jnl of Applied Mathematics, 9 (1998), 417-445.  doi: 10.1017/S0956792598003520.  Google Scholar [7] L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 79 (2013), 12–27, arXiv: 1206.6738. doi: 10.1016/j.na.2012.11.010.  Google Scholar [8] O. Cârjă, A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions, Nonlinear Anal. TMA, 113 (2015), 190-208.  doi: 10.1016/j.na.2014.10.003.  Google Scholar [9] X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differential Equations, 206 (2004), 399-437.  doi: 10.1016/j.jde.2004.05.017.  Google Scholar [10] L. Cherfils, S. Gatti and A. Miranville, Long time behavior to the Caginalp system with singular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 11 (2012), 2261-2290.  doi: 10.3934/cpaa.2012.11.2261.  Google Scholar [11] K. Chrysafinos, Discontinuous Time-Stepping Schemes for the Allen-Cahn Equations and Applications to Optimal Control, Paper presented at the Conference on 'Advances in scientific computing and applied mathematics', Las Vegas, Nevada, October 2015. Google Scholar [12] X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math., 94 (2003), 33-65.  doi: 10.1007/s00211-002-0413-1.  Google Scholar [13] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Clarendon, Oxford, 1995. Google Scholar [14] C. Gal, M. Grasselli and A. Miranville, Non-isothermal Allen-Cahn equations with coupled dynamic boundary conditions, Nonlinear Phenomena with Energy Dissipation, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 29 (2008), 117–139.  Google Scholar [15] Gh. Iorga, C. Moroşanu and S. C. Cocindău, Numerical simulation of the solid region via phase field transition system, Metalurgia International, 8 (2008), 91-95.   Google Scholar [16] Gh. Iorga, C. Moroşanu and I. Tofan, Numerical simulation of the thickness accretions in the secondary cooling zone of a continuous casting machine, Metalurgia International, 14 (2009), 72-75.   Google Scholar [17] H. Israel, Long time behavior of an Allen-Cahn type equation with singular potential and dynamic boundary conditions, Journal of Applied Analysis and Computation, 2 (2012), 29-56.   Google Scholar [18] N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal. TMA, 22 (1994), 1163-1180.  doi: 10.1016/0362-546X(94)90235-6.  Google Scholar [19] O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasi-Linear Equations of Parabolic Type, Prov. Amer. Math. Soc., 1968. Google Scholar [20] J. L. Lions, Control of Distributed Singular Systems, Gauthier-Villars, Paris, 1985. Google Scholar [21] A. Miranville, Existence of solutions for a one-dimensional Allen-Cahn equation, J. Appl. Anal. Comput., 3 (2013), 265-277.   Google Scholar [22] A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions, Appl. Math. Modell., 40 (2016), 192-207.  doi: 10.1016/j.apm.2015.04.039.  Google Scholar [23] A. Miranville and C. Moroşanu, Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions, Discrete and Continuous Dynamical Systems Series S, 9 (2016), 537-556.  doi: 10.3934/dcdss.2016011.  Google Scholar [24] C. Moroşanu, Approximation and numerical results for phase field system by a fractional step scheme, Rev. Anal. Numér. Théor. Approx., 25 (1996), 137-151.   Google Scholar [25] C. Moroşanu, Approximation of the phase-field transition system via fractional steps method, Numer. Funct. Anal. and Optimiz., 18 (1997), 623-648.  doi: 10.1080/01630569708816782.  Google Scholar [26] C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods, Bentham Science Publishers, 2012. doi: 10.2174/97816080535061120101.  Google Scholar [27] C. Moroşanu, Qualitative and quantitative analysis for a nonlinear reaction-diffusion equation, ROMAI J., 12 (2016), 85–113. https://rj.romai.ro/arhiva/2016/2/Morosanu.pdf  Google Scholar [28] C. Moroşanu and A. Croitoru, Analysis of an iterative scheme of fractional steps type associated to the phase-field equation endowed with a general nonlinearity and Cauchy-Neumann boundary conditions, J. Math. Anal. Appl., 425 (2015), 1225-1239.  doi: 10.1016/j.jmaa.2015.01.033.  Google Scholar [29] C. Moroşanu and A.-M. Moşneagu, On the numerical approximation of the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions. Case 1D, ROMAI J., 9 (2013), 91–110, https://rj.romai.ro/arhiva/2013/1/Morosanu,Mosneagu.pdf  Google Scholar [30] C. Moroşanu and D. Motreanu, A generalized phase field system, J. Math. Anal. Appl., 237 (1999), 515-540.  doi: 10.1006/jmaa.1999.6467.  Google Scholar [31] C. Moroşanu and D. Motreanu, The phase field system with a general nonlinearity, International Journal of Differential Equations and Applications, 1 (2000), 187-204.   Google Scholar [32] C. Moroşanu, S. Pavăl and C. Trenchea, Analysis of stability and errors of three methods associated to the nonlinear reaction-diffusion equation supplied with homogeneous Neumann boundary conditions, Journal of Applied Analysis and Computation, 7 (2017), 1-19.  doi: 10.11948/2017001.  Google Scholar [33] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar [34] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for kinetics of phase transitions, Phys. D., 43 (1990), 44-62.  doi: 10.1016/0167-2789(90)90015-H.  Google Scholar [35] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences, Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [36] X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 1057-1070.  doi: 10.3934/dcdsb.2009.11.1057.  Google Scholar [37] J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM J. Sci. Comput., 31 (2009), 3042-3063.  doi: 10.1137/080738398.  Google Scholar
Numerical stability: $V^i$ at different levels of time
Errors $\|v_e-V_j^M\|_\infty$ of the Newton, the linearized and the fractional steps methods: (10)-(11)
Errors $\|v_e-V_j^M\|_\infty$ of the Newton, the linearized and the fractional steps methods: (10)-(11)
Errors $\|v_e-V_j^M\|_\infty$ of the Newton, the linearized and the fractional steps methods: (10)-(11)
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