×
Wu, Fengyan; Cheng, Xiujun; Li, Dongfang; Duan, Jinqiao

A two-level linearized compact ADI scheme for two-dimensional nonlinear reaction-diffusion equations. (English) Zbl 1415.65202

Comput. Math. Appl. 75, No. 8, 2835-2850 (2018).
Summary: A novel two-level linearized compact alternating direction implicit (ADI) scheme is proposed for solving two-dimensional nonlinear reaction-diffusion equations. The computational cost is reduced by use of the Newton linearized method and the ADI method. The existence and uniqueness of the numerical solutions are proved. Moreover, the error estimates in \(H^1\) and \(L^\infty\) norms are presented. Numerical examples are given to illustrate our theoretical results.

Cited in 29 Documents

MSC:

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
35K20 Initial-boundary value problems for second-order parabolic equations
35K58 Semilinear parabolic equations

Keywords:

nonlinear reaction-diffusion equations; compact ADI scheme; Newton linearized approximation; convergence; stability
Cite Review PDF
Full Text: DOI

References:

[1] Li, N.; Steiner, J.; Tang, S., Convergence and stability analysis of an explicit finite difference method for 2-dimensional reaction-diffusion equations, J. Aust. Math. Soc. Ser. B., 36, 234-241 (1994) · Zbl 0827.65088
[2] Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations (1997), Springer-Verlag: Springer-Verlag New York
[3] Araújo, A.; Barbeiro, S.; Serranho, P., Convergence of finite difference schemes for nonlinear complex reaction-diffusion processes, SIAM J. Numer. Anal., 53, 228-250 (2015) · Zbl 1327.65167
[4] Li, D.; Zhang, C., Split Newton iterative algorithm and its application, Appl. Math. Comput., 217, 2260-2265 (2010) · Zbl 1202.65066
[5] Li, S., Numerical analysis for fourth-order compact conservative difference scheme to solve the 3D Rosenau-RLW equation, Comp. Math. Appl., 72, 2388-2407 (2016) · Zbl 1368.65141
[6] Wang, Y.; Wang, J., A higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equaionts, Comput. Math. Appl., 62, 2434-2451 (2011) · Zbl 1231.65148
[7] Fernandes, R. I.; Bialecki, B.; Fairweather, G., An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems on evolving domains, J. Comput. Phys., 299, 561-580 (2015) · Zbl 1352.65393
[8] Li, B.; Gao, H.; Sun, W., Unconditionally optimal error estimates of a Crank-Nicolson Galerkin method for the nonlinear thermistor equations, SIAM J. Numer. Anal., 53, 933-954 (2014) · Zbl 1298.65160
[9] Li, D.; Wang, J., Unconditionally optimal error analysis of Crank-Nicolson Galerkin FEMs for a strongly nonlinear parabolic system, J. Sci. Comput., 72, 892-915 (2017) · Zbl 1377.65118
[10] Wang, J.; Sun, W., Optimal error analysis of Crank-Nicolson schemes for a coupled nonlinear Schrödinger system in 3D, J. Comput. Appl. Math., 317, 685-699 (2017) · Zbl 1357.65148
[11] Wu, H., An alternating direction implicit difference scheme for two dimensional semilinear reaction-diffusion equation, Math. Numer. Sinica, 30, 4, 349-360 (2008) · Zbl 1174.65471
[12] Zhao, X.; Sun, Z. Z.; Hao, Z., A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation, SIAM. J. Sci. Comput., 36, A2865-A2886 (2014) · Zbl 1328.65187
[13] Akrivis, G.; Li, B., Maximum norm analysis of implicit-explicit backward difference formulae for nonlinear parabolic equations, IMA J Numer. Math. (2017)
[14] Ascher, U. M.; Ruuth, S. J.; Wetton, B., Implicit-explicit methods for time dependent partial differential equations, SIMA J. Numer. Anal., 32, 797-823 (1995) · Zbl 0841.65081
[15] Dupont, T.; Fairweather, G.; Johnson, J. P., Three-level Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 11, 392-410 (1974) · Zbl 0313.65107
[16] Li, D.; Wang, J.; Zhang, J., Unconditionally convergent \(L 1\)-Galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM J. Sci. Comput., 39, 6, A3067-A3088 (2017) · Zbl 1379.65079
[17] Owolabi, K. M.; Patidar, K. C., Higher-order time-stepping methods for time-dependent reaction-diffusion equations arising in biology, Appl. Math. Comput., 240, 30-50 (2014) · Zbl 1334.65136
[18] Owolabi, K. M., Robust IMEX schemes for solving two-dimensional reaction-diffusion models, Int. J. Nonl. Sci. Numer. Simu., 15, 271-284 (2015) · Zbl 1401.65100
[19] Zhang, Q.; Li, D.; Zhang, C., Multistep finite difference schemes for the variable coefficient delay parabolic equations, J. Difference Equ. Appl., 22, 1-21 (2016)
[20] Liao, H.; Sun, Z., Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations, Numer. Meth. Part. Diff. Equ., 26, 37-60 (2010) · Zbl 1196.65154
[21] Peaceman, D. W.; Rachford, H. H., The numerical solution of elliptic and parabolic differential equations, J. Soc. Ind. Appl. Math., 3, 28-41 (1955) · Zbl 0067.35801
[22] Zhao, X.; Sun, Z. Z., Compact Crank-Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium, J. Sci. Comput., 62, 747-771 (2015) · Zbl 1319.65084
[23] Deng, D.; Zhang, C., Analysis and application of a compact multistep ADI solver for a class of nonlinear viscous wave equations, Appl. Math. Model., 39, 1033-1049 (2015) · Zbl 1432.65119
[24] Xiao, A.; Zhang, G.; Yi, X., Two classes of implicit-explicit multistep methods for nonlinear stiff initial-value problems, Appl. Math. Comput., 247, 47-60 (2014) · Zbl 1338.65179
[25] Sun, Z.; Zhang, Z., A linearized compact difference scheme for a class of nonlinear delay partial differential equations, Appl. Math. Model., 37, 3, 742-752 (2013) · Zbl 1352.65270
[26] Zhou, H.; Wu, Y.; Tian, W., Extrapolation algorithm of compact ADI approximation for two-dimensional parabolic equation, Appl. Math. Comput., 219, 2875-2884 (2012) · Zbl 1309.65102
[27] Danilov, V. G.; Maslov, V. P.; Volosov, K. A., Mathematical Modelling of Heat and Mass Transfer Processes (1995), Kluwer: Kluwer Dordrecht · Zbl 0839.35001
[28] Li, D.; Zhang, C.; Wang, W.; Zhang, Y., Implicit-explicit predictor-corrector schemes for nonlinear parabolic differential equations, Appl. Math. Model., 35, 6, 2711-2722 (2011) · Zbl 1219.65098
[29] Sachdev, P. L.; Valli, A., Nonlinear Diffusive Waves (2009), Cambridge University Press
[30] Chafee, N.; Infante, E. F., A bifurcation problem for a nonlinear partial differential equation of parabolic type, Appl. Anal., 4, 1, 17-37 (1974) · Zbl 0296.35046
[31] Constantin, P.; Foias, C.; Nicolaenko, B., Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations (2012), Springer Science & Business Media · Zbl 0683.58002
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.