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An adaptive moving grid method for a system of singularly perturbed initial value problems. (English) Zbl 1296.65102

Summary: A system of first-order singularly perturbed initial value problems is considered. The system is discretized by a backward Euler difference scheme for which a priori error analysis in the maximum norm is constructed. It is shown from the a priori error bound that there exists a mesh with \(N\) subintervals that gives optimal error bound of \(O(N^{- 1})\) which is robust with respect to the perturbation parameters. A partly heuristic argument based on a priori error analysis leads to a suitable monitor function. Based on an a posteriori error bound, a first-order rate of convergence, independent of all perturbation parameters, is established. A linear and a nonlinear example are tested, and the numerical results are provided to demonstrate the effectiveness of our adaptive moving grid method.

MSC:

65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34E15 Singular perturbations for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
34A30 Linear ordinary differential equations and systems
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[1] Bawa, R. K.; Natesan, S., A computational method for self-adjoint singular perturbation problems using quintic spline, Comput. Math. Appl., 50, 1371-1382 (2005) · Zbl 1084.65070
[2] Natesan, S.; Jayakumar, J.; Vigo-Aguiar, J., Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers, J. Comput. Appl. Math., 158, 121-134 (2003) · Zbl 1033.65061
[3] Vigo-Aguiar, J.; Natesan, S., An efficient numerical method for singular perturbation problems, J. Comput. Appl. Math., 192, 132-141 (2006) · Zbl 1095.65068
[4] Ramos, H.; Vigo-Aguiar, J.; Natesan, S., Numerical solution of nonlinear singularly perturbed problems on nonuniform meshes by using a non-standard algorithm, J. Math. Chem., 48, 38-54 (2010) · Zbl 1304.65180
[5] Natesan, S.; Vigo-Aguiar, J.; Ramanujam, N., A numerical algorithm for singular perturbation problems exhibiting weak boundary layers, Comput. Math. Appl., 45, 469-479 (2003) · Zbl 1036.65065
[6] Athanasios, A. C., Approximation of Large-Scale Dynamical Systems (2005), SIAM: SIAM Philadelphia · Zbl 1112.93002
[7] Meenakshi, P. M.; Valarmathi, S.; Miller, J. J.H., Solving a partially singularly perturbed initial value problem on Shishkin meshes, Appl. Math. Comput., 215, 3170-3180 (2010) · Zbl 1208.65099
[8] Valarmathi, S.; Miller, J. J.H., A parameter-uniform finite difference method for singulary perturbed linear dynamical systems, Int. J. Numer. Anal. Model., 7, 535-548 (2010) · Zbl 1198.65123
[9] Kumar, S.; Kumar, M., Parameter-robust numerical method for a system of singulary perturbed initial value problems, Numer. Algorithms, 59, 185-195 (2012) · Zbl 1245.65101
[10] Rao, S. C.S.; Kumar, S., Second order global uniformly convergent numerical method for a coupled system of singularly perturbed initial value problems, Appl. Math. Comput., 219, 3740-3753 (2012) · Zbl 1311.65101
[11] Cen, Z. D.; Xu, A. M.; Le, A. B., A second-order hybird finite difference scheme for a system of singularly perturbed initial value problems, J. Comput. Appl. Math., 234, 3445-3457 (2010) · Zbl 1197.65095
[12] Chen, Y., Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid, Adv. Comput. Math., 24, 197-212 (2006) · Zbl 1095.65065
[13] Kopteva, N., Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39, 423-441 (2001) · Zbl 1003.65091
[14] Kopteva, N.; Stynes, M., A robust adaptive method for quasi-linear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39, 1446-1467 (2001) · Zbl 1012.65076
[15] Chen, Y., Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution, J. Comput. Appl. Math., 159, 25-34 (2003) · Zbl 1044.65066
[16] Mackenzie, J., Uniform convergence analysis of an upwind finite difference approximation of a convection-diffusion bondary value problem on an adaptive gird, IMA J. Numer. Anal., 19, 233-249 (1999) · Zbl 0929.65047
[17] Gowrisankar, S.; Natesan, S., The parameter uniform numerical method for singularly perturbed parabolic reaction-diffusion problems on equidistributed grids, Appl. Math. Lett., 26, 1053-1060 (2013) · Zbl 1311.65111
[18] Farrell, P. A.; Hegarty, A. F.; Miller, J. J.H.; O’Riordan, E.; Shishkin, G. I., Robust Computational Techniques for Boundary Layers (2000), Chapman & Hall/ CRC: Chapman & Hall/ CRC Boca Raton · Zbl 0964.65083
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