• Previous Article
    Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory
  • DCDS-S Home
  • This Issue
  • Next Article
    The Stokes problem in fractal domains: Asymptotic behaviour of the solutions
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 and 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.

[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.

[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.

[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.

[5]

T. BenincasaA. 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.

[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.

[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.

[8]

O. CârjăA. 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.

[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.

[10]

L. CherfilsS. 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.

[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.

[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.

[13]

I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Clarendon, Oxford, 1995.

[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.

[15]

Gh. IorgaC. Moroşanu and S. C. Cocindău, Numerical simulation of the solid region via phase field transition system, Metalurgia International, 8 (2008), 91-95. 

[16]

Gh. IorgaC. 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. 

[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. 

[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.

[19]

O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasi-Linear Equations of Parabolic Type, Prov. Amer. Math. Soc., 1968.

[20]

J. L. Lions, Control of Distributed Singular Systems, Gauthier-Villars, Paris, 1985.

[21]

A. Miranville, Existence of solutions for a one-dimensional Allen-Cahn equation, J. Appl. Anal. Comput., 3 (2013), 265-277. 

[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.

[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.

[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. 

[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.

[26]

C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods, Bentham Science Publishers, 2012. doi: 10.2174/97816080535061120101.

[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

[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.

[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

[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.

[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. 

[32]

C. MoroşanuS. 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.

[33] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 
[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

T. BenincasaA. 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.

[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.

[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.

[8]

O. CârjăA. 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.

[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.

[10]

L. CherfilsS. 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.

[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.

[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.

[13]

I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Clarendon, Oxford, 1995.

[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.

[15]

Gh. IorgaC. Moroşanu and S. C. Cocindău, Numerical simulation of the solid region via phase field transition system, Metalurgia International, 8 (2008), 91-95. 

[16]

Gh. IorgaC. 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. 

[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. 

[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.

[19]

O. A. Ladyzhenskaya, B. A. Solonnikov and N. N. Uraltzava, Linear and Quasi-Linear Equations of Parabolic Type, Prov. Amer. Math. Soc., 1968.

[20]

J. L. Lions, Control of Distributed Singular Systems, Gauthier-Villars, Paris, 1985.

[21]

A. Miranville, Existence of solutions for a one-dimensional Allen-Cahn equation, J. Appl. Anal. Comput., 3 (2013), 265-277. 

[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.

[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.

[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. 

[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.

[26]

C. Moroşanu, Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods, Bentham Science Publishers, 2012. doi: 10.2174/97816080535061120101.

[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

[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.

[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

[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.

[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. 

[32]

C. MoroşanuS. 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.

[33] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 
[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.

[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.

[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.

[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.

Figure 1.  Numerical stability: $ V^i $ at different levels of time
Figure 2.  Errors $ \|v_e-V_j^M\|_\infty $ of the Newton, the linearized and the fractional steps methods: (10)-(11)
Figure 3.  Errors $ \|v_e-V_j^M\|_\infty $ of the Newton, the linearized and the fractional steps methods: (10)-(11)
Figure 4.  Errors $ \|v_e-V_j^M\|_\infty $ of the Newton, the linearized and the fractional steps methods: (10)-(11)
[1]

Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515

[2]

Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269

[3]

Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289

[4]

Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402

[5]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319

[6]

Ya-Xiang Yuan. Recent advances in numerical methods for nonlinear equations and nonlinear least squares. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 15-34. doi: 10.3934/naco.2011.1.15

[7]

Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561

[8]

Shalva Amiranashvili, Raimondas  Čiegis, Mindaugas Radziunas. Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinetic and Related Models, 2015, 8 (2) : 215-234. doi: 10.3934/krm.2015.8.215

[9]

Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304

[10]

Hong Wang, Aijie Cheng, Kaixin Wang. Fast finite volume methods for space-fractional diffusion equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1427-1441. doi: 10.3934/dcdsb.2015.20.1427

[11]

Jinye Shen, Xian-Ming Gu. Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1179-1207. doi: 10.3934/dcdsb.2021086

[12]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133

[13]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581

[14]

Jitraj Saha, Nilima Das, Jitendra Kumar, Andreas Bück. Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinetic and Related Models, 2019, 12 (1) : 79-103. doi: 10.3934/krm.2019004

[15]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[16]

Asif Yokus, Mehmet Yavuz. Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2591-2606. doi: 10.3934/dcdss.2020258

[17]

Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155

[18]

Caojin Zhang, George Yin, Qing Zhang, Le Yi Wang. Pollution control for switching diffusion models: Approximation methods and numerical results. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3667-3687. doi: 10.3934/dcdsb.2018310

[19]

Filipa Caetano, Martin J. Gander, Laurence Halpern, Jérémie Szeftel. Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations. Networks and Heterogeneous Media, 2010, 5 (3) : 487-505. doi: 10.3934/nhm.2010.5.487

[20]

Emmanuel Frénod. Homogenization-based numerical methods. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : i-ix. doi: 10.3934/dcdss.201605i

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (182)
  • HTML views (416)
  • Cited by (0)

Other articles
by authors

[Back to Top]