Implicit–explicit multistep finite-element methods for nonlinear convection-diffusion-reaction equations with time delay

References

• G. Akrivis and C. Lubich, Fully implicit, linearly implicit and implicit–explicit backward difference formulae for quasi-linear parabolic equations, Numer. Math. 131(4) (2015), pp. 713–735. doi: 10.1007/s00211-015-0702-0
   Web of Science ®Google Scholar
• G. Akrivis, B. Li, and C. Lubich, Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations, Math. Comput. 86(306) (2017), pp. 1527–1552. doi: 10.1090/mcom/3228
   Web of Science ®Google Scholar
• U.M. Ascher, S.J. Ruuth, and B.T.R. Wetton, Implicit–explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal. 32 (1995), pp. 797–823. doi: 10.1137/0732037
   Web of Science ®Google Scholar
• B. Basse, B.C. Baguley, E.S. Marshall, W.R. Joseph, B. van Brunt, G. Wake, and D.J.N. Wall, A mathematical model for analysis of the cell cycle in cell lines derived from human tumors, J. Math. Biol. 47 (2003), pp. 295–312. doi: 10.1007/s00285-003-0203-0
   PubMed Web of Science ®Google Scholar
• A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003.
   Google Scholar
• A.H. Bhrawy, M.A. Abdelkawy, and F. Mallawi, An accurate Chebyshev pseudospectral scheme for multi-dimensional parabolic problems with time delays, Bound. Value Probl. 2015 (2015), pp. 103–100. doi: 10.1186/s13661-015-0364-y
   Google Scholar
• W. Chen, Implicit–explicit multistep finite element methods for nonlinear convection-diffusion problems, Numer. Methods Partial Diff Equ. 17 (2001), pp. 93–104. doi: 10.1002/1098-2426(200103)17:2<93::AID-NUM1>3.0.CO;2-B
   Web of Science ®Google Scholar
• W. Chen, Implicit–explicit multistep finite element-mixed finite element methods for the transient behavior of a semiconductor device, Acta Math. Sci. 237 (2003), pp. 386–398.
   Google Scholar
• P.G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM Classics in Applied Mathematics, Vol. 40, SIAM, Philadelphia, 2002.
   Google Scholar
• Z. Jackiewicz, H. Liu, B. Li, and Y. Kuang, Numerical simulations of traveling wave solutions in a drift paradox inspired diffusive delay population model, Math. Comput. Simul. 96(2) (2014), pp. 95–103. doi: 10.1016/j.matcom.2012.06.004
   Google Scholar
• O. Lakkis, A. Madzvamuse, and C. Venkataraman, Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems on evolving domains, SIAM J. Numer. Anal. 51(4) (2013), pp. 2309–2330. doi: 10.1137/120880112
   Web of Science ®Google Scholar
• B. Liu and C. Zhang, A spectral Galerkin method for nonlinear delay convection-diffusion-reaction equations, Comput. Math. Appl. 69(8) (2015), pp. 709–724. doi: 10.1016/j.camwa.2015.02.027
   Web of Science ®Google Scholar
• C.V. Pao, Numerical methods for systems of nonlinear parabolic equations with time delays, J. Math. Anal. Appl. 240(1) (1999), pp. 249–279. doi: 10.1006/jmaa.1999.6619
   Web of Science ®Google Scholar
• H.J. Wang, C.-W. Shu, and Q. Zhang, Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems, SIAM J. Numer. Anal. 53(1) (2015), pp. 206–227. doi: 10.1137/140956750
   Web of Science ®Google Scholar
• H.J. Wang, C.W. Shu, and Q. Zhang, Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for nonlinear convection-diffusion problems, Appl. Math. Comput. 272 (2016), pp. 237–258.
   Web of Science ®Google Scholar
• J.H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996.
   Google Scholar
• A.G. Xiao, G.G. Zhang, and X. Yi, Two classes of implicit–explicit multistep methods for nonlinear stiff initial-value problems, Appl. Math. Comput. 247 (2014), pp. 47–60.
   Web of Science ®Google Scholar
• A.G. Xiao, G.G. Zhang, and J. Zhou, Implicit-explicit time discretization coupled with finite element methods for delayed predator-prey competition reaction-diffusion system, Comput. Math. Appl. 71(10) (2016), pp. 2106–2123. doi: 10.1016/j.camwa.2016.04.003
   Web of Science ®Google Scholar
• G.G. Zhang and A.G. Xiao, Stability and convergence analysis of implicit-explicit one-leg methods for stiff delay differential equations, Int. J. Comput. Math. 93(11) (2016), pp. 1964–1983. doi: 10.1080/00207160.2015.1080359
   Web of Science ®Google Scholar
• Q. Zhang and C. Zhang, A new linearized compact multisplitting scheme for the nonlinear convection-reaction-diffusion equations with delay, Commun. Nonlinear Sci. Numer. Simul. 18(18) (2013), pp. 3278–3288. doi: 10.1016/j.cnsns.2013.05.018
   Google Scholar
• K. Zhang, J.C.F. Wong, and R. Zhang, Second-order implicit–explicit scheme for the Gray-Scott model, J. Comput. Appl. Math. 213 (2008), pp. 559–581. doi: 10.1016/j.cam.2007.01.038
   Web of Science ®Google Scholar