An improved uniformly convergent scheme in space for 1D parabolic reaction–diffusion systems

Bakhvalov, 1969, On the optimization of methods for boundary-value problems with boundary layers, J. Numer. Methods Math. Phys., 9, 841 Barenblatt, 1960, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24, 1286, 10.1016/0021-8928(60)90107-6 Clavero, 2012, A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction–diffusion problems, Appl. Math. Comput., 218, 5067, 10.1016/j.amc.2011.10.072 Clavero, 2009, High order schemes for reaction–diffusion singularly perturbed systems, Lect. Notes Comput. Sci. Eng., 69, 107, 10.1007/978-3-642-00605-0_7 Clavero, 2010, An almost third order finite difference scheme for singularly perturbed reaction–diffusion systems, J. Comput. Appl. Math., 234, 2501, 10.1016/j.cam.2010.03.011 Clavero, 2010, Second order uniform approximations for the solution of time dependent singularly perturbed reaction–diffusion systems, Int. J. Numer. Anal. Mod., 7, 428 P.A. Farrell, A. Hegarty, On the determination of the order of uniform convergence, in: Proceedings of IMACS’91, vol. 2, 1991, pp. 501–502. Farrell, 2000 Gartland, 1987, Uniform high-order difference schemes for a singularly perturbed two point boundary value problem, Math. Comput., 48, 551, 10.1090/S0025-5718-1987-0878690-0 Gracia, 2007, A uniformly convergent numerical scheme for a system of reaction–diffusion equations, J. Comput. Appl. Math., 206, 1, 10.1016/j.cam.2006.06.005 Herceg, 1990, Uniform fourth order difference scheme for a singular perturbation problem, Numer. Math., 56, 675, 10.1007/BF01405196 Ladyzhenskaya, 1968 Linss, 2003, An improved error estimate for a numerical method for a system of coupled singularly perturbed reaction–diffusion equations, Comput. Methods Appl. Math., 3, 417, 10.2478/cmam-2003-0027 Linss, 2009, Numerical solution of systems of singularly perturbed differential equations, Comput. Methods Appl. Math., 9, 165, 10.2478/cmam-2009-0010 Madden, 2003, A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction–diffusion problems, IMA J. Numer. Anal., 23, 627, 10.1093/imanum/23.4.627 N. Madden, M. Stynes, G. Thomas, On the application of robust numerical methods to a complete-flow wave-current model, Proceedings of BAIL 2004, 2004. Matthews, 2002, A numerical method for a system of singularly perturbed reaction–diffusion equations, J. Comput. Appl. Math., 145, 151, 10.1016/S0377-0427(01)00541-6 Protter, 1967 Roos, 2008, Robust numerical methods for singularly perturbed differential equations, 24 Sekhara Rao, 2013, An almost fourth order parameter-robust numerical method for a linear system of (M⩾2) coupled singularly perturbed reaction–diffusion problems, Int. J. Numer. Anal. Model., 10, 603 Shishkin, 1995, Mesh approximation of singularly perturbed boundary-value problems for systems of elliptic and parabolic equations, Comput. Math. Math. Phys., 35, 429 G.P. Thomas, Towards an improved turbulence model for wave–current interactions, in: Second Annual Report to EU MAST-III Project The Kinematics and Dynamics of Wave–Current Interactions, 1998. Vulanović, 2001, A higher-order scheme for quasilinear boundary value problems with two small parameters, Computing, 67, 287, 10.1007/s006070170002 Vulanović, 2004, An almost sixth-order finite-difference method for semilinear singular perturbation problems, Comput. Methods Appl. Math., 4, 368, 10.2478/cmam-2004-0020 Xenophontos, 2010, On the finite element approximation of systems of reaction–diffusion equations by p/hp methods, J. Comput. Math., 28, 386, 10.4208/jcm.2009.10-m2636