Abstract
This paper discusses the Richardson extrapolation technique of an upwind finite difference scheme on a modified Bakhvalov-type mesh (named Vulanović–Bakhvalov mesh) for a model singularly perturbed convection–diffusion problem in two dimensions. We give a rigorous convergence analysis for the Richardson extrapolation method, which is shown that the use of Richardson extrapolation technique improves the \(\varepsilon \)-uniform accuracy of the presented scheme in the discrete maximum norm from \(O\left( N^{-1}\right) \) to \(O\left( N^{-2}\right) \), where N is the number of mesh-intervals. Then we also propose a new numerical method to obtain the optimization mesh parameters of the Vulanović–Bakhvalov mesh. Finally some numerical experiments are provided to validate our theoretical results.
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References
Alhumaizi K (2007) Flux-limiting solution techniques for simulation of reaction–diffusion–convection system. Commun Nonlinear Sci Numer Simul 12(6):953–965
Bakhvalov NS (1969) The optimization of methods of solving boundary value problems with a boundary layer. USSR Comp Math Math Phys 9:139–166
Durán RG, Lombardi AL, Prieto MI (2012) Superconvergence for finite element approximation of a convection–diffusion equation using graded meshes. IMA J Numer Anal 32(2):511–533
Ewing RE (ed) (1983) The mathematics of reservoir simulation, Frontiers in Applied Mathematics, vol 1. Society for Industrial and Applied Mathematics, Philadelphia
Franz S, Roos HG (2011) The capriciousness of numerical methods for singular perturbations. SIAM Rev 53:157–173
Kopteva N (2003) Error expansion for an upwind scheme applied to a two-dimensional convection–diffusion problem. SIAM J Numer Anal 41(5):1851–1869
Kreiss H-O, Lorenz J (1989) Initial-boundary value problems and the Navier–Stokes equations, Pure and Applied Mathematics, vol 136. Academic Press Inc, Boston
Linß T (1998) Layer-adapted meshes for reaction–convection-diffusion problems. Springer, Berlin
Linß T (1999) An upwind difference scheme on a novel Shishkin-type mesh for a linear convection–diffusion problem. J Comput Appl Math 110(1):93–104
Linß T, Stynes M (1999) A hybrid difference scheme on a Shishkin mesh for linear convection–diffusion problems. Appl Numer Math 31(3):255–270
Linß T, Stynes M (2001) Asymptotic anaysis and Shishkin-type decomposition for an elliptic convection–diffusion problem. J Math Anal Appl 262:604–632
Linß T, Stynes M (2001) Asymptotic analysis and shishkin-type decomposition for an elliptic convection-diffusion problem. J Math Anal Appl 261(2):604–632
Liu XW, Martin S, Zhang J (2017) Supercloseness of edge stabilization on Shishkin rectangular meshes for convection–diffusion problems with exponential layers. IIMA J Numer Anal 38:2105–2122
Mukherjee K, Natesan S (2011) Richardson extrapolation technique for singularly perturbed parabolic convection-diffusion problems. Computing 92(1):1–32
Nhan TA, Mai VQ (2021) On Bakhvalov-type meshes for a linear convection-diffusion problem in 2D. Math Commun 26(2):121–130
Nhan TA, Vulanović R (2018) A note on a generalized Shishkin-type mesh. Novi Sad J Math 48(2):141–150
Nhan TA, Vulanović R (2019) Analysis of the truncation error and barrier-function technique for a Bakhvalov-type mesh. Electron T Numer Anal 51:315–330
Nhan TA, Vulanović R (2020) The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems. Numer Algor 87:203–221
Olsson DM, Nelson LS (1975) The Nelder-Mead simplex procedure for function minimization. Technometrics 17:45–51
Roos HG (2019) Layer-adapted meshes: milestones in 50 years of histroy. Appl. Math. arXiv:1909.08273v1
Roos HG, Stynes M (2015) Some open questions in the numerical analysis of singularly perturbed differential equations. Comput Meth Appl Mat 15(4):531–550
Roos HG, Stynes M, Tobiska L (2008) Robust numerical methods for singularly perturbed differential equations: convection–diffusion reaction and flow problem. Springer Science & Business Media, Berlin
Singh MK, Natesan S (2018) Richardson extrapolation technique for singularly perturbed system of parabolic partial differential equations with exponential boundary layers. Appl Math Comput 333:254–275
Vulanović R (1983) On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh. Univ. u Novom Sadu Zb. Rad. Prir. Mat. Fak. Ser. Mat. 13, 187-201
Zhang J, Lv YH (2021) Finite element method for singularly perturbed problems with two parameters on a Bakhvalov-type mesh in 2D. Numer Algor. https://doi.org/10.1007/s11075-021-01194-7
Acknowledgements
This work is supported by the key project of Natural Science Foundation of Guangxi Province (AD20238065), the Natural Science Foundation of Guangxi Province (2020GXNSFAA159010), and the National Science Foundation of China (12261062).
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Communicated by Justin Wan.
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Liao, Y., Liu, LB., Xu, L. et al. Richardson extrapolation method for 2D-SPP on VB mesh singularly perturbed convection–diffusion problem on a Vulanović–Bakhvalov mesh. Comp. Appl. Math. 42, 117 (2023). https://doi.org/10.1007/s40314-023-02260-7
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DOI: https://doi.org/10.1007/s40314-023-02260-7