A high-order nonuniform compact difference scheme coupled with adaptive mesh method for solving singularly perturbed convection-diffusion equations

Tao Wang; Jinhua Ma; Qiong Chen; Tao Wang; Jinhua Ma; Qiong Chen

doi:10.3934/math.2026092

AIMS Mathematics

2026, Volume 11, Issue 1: 2279-2312. doi: 10.3934/math.2026092

Previous Article Next Article

Research article

A high-order nonuniform compact difference scheme coupled with adaptive mesh method for solving singularly perturbed convection-diffusion equations

1.
School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China
2.
Ningxia Collaborative Innovation Center for Scientific Computing and Intelligent Information Processing, Yinchuan, 750021, China

Received: 10 October 2025 Revised: 02 January 2026 Accepted: 12 January 2026 Published: 23 January 2026
MSC : 65D25, 65N06, 65Y04, 65Z05

A high-order nonuniform compact difference scheme coupled with an adaptive mesh method for solving singularly perturbed convection-diffusion equations is proposed. The core idea of the proposed method lies in constructing a nonuniform grid scheme with the same fourth-order accuracy at both inner and boundary points, and this construction is based on an adaptive mesh method. This approach effectively eliminates numerical oscillation near the boundaries, a common challenge in singularly perturbed problems. Specifically, we first formulate the fourth-order compact scheme on nonuniform grids, then integrate it with the adaptive mesh method and elaborate on the corresponding numerical implementation procedure. Finally, numerical experiments are conducted against exact solutions, with the proposed scheme further compared against three benchmark methods: the fourth-order compact scheme on uniform grids, the identical scheme on adaptive nonuniform grids, and the other established methods reported in existing literature. Results of all test cases demonstrate that the proposed scheme generates accurate and stable numerical solutions and exhibits enhanced resolution for singularly perturbed convection-diffusion problems.
- singularly perturbed convection-diffusion equation,
- adaptive mesh method,
- consistent fourth-order nonuniform scheme,
- boundary layer
Citation: Tao Wang, Jinhua Ma, Qiong Chen. A high-order nonuniform compact difference scheme coupled with adaptive mesh method for solving singularly perturbed convection-diffusion equations[J]. AIMS Mathematics, 2026, 11(1): 2279-2312. doi: 10.3934/math.2026092

Related Papers:

Abstract

A high-order nonuniform compact difference scheme coupled with an adaptive mesh method for solving singularly perturbed convection-diffusion equations is proposed. The core idea of the proposed method lies in constructing a nonuniform grid scheme with the same fourth-order accuracy at both inner and boundary points, and this construction is based on an adaptive mesh method. This approach effectively eliminates numerical oscillation near the boundaries, a common challenge in singularly perturbed problems. Specifically, we first formulate the fourth-order compact scheme on nonuniform grids, then integrate it with the adaptive mesh method and elaborate on the corresponding numerical implementation procedure. Finally, numerical experiments are conducted against exact solutions, with the proposed scheme further compared against three benchmark methods: the fourth-order compact scheme on uniform grids, the identical scheme on adaptive nonuniform grids, and the other established methods reported in existing literature. Results of all test cases demonstrate that the proposed scheme generates accurate and stable numerical solutions and exhibits enhanced resolution for singularly perturbed convection-diffusion problems.

References

[1]	K. Kumar, P. C. Pidola, A new stable finite difference scheme and its error analysis for two-dimensional singularly perturbed convection–diffusion equations, Numer. Methods Partial Differential Equations, 38 (2020), 1215–1231. https://doi.org/10.1002/num.22732 doi: 10.1002/num.22732
[2]	S. Elango, Second order singularly perturbed delay differential equations with non-local boundary condition, J. Comput. Appl. Math., 417 (2023), 114498. https://doi.org/10.1016/j.cam.2022.114498 doi: 10.1016/j.cam.2022.114498
[3]	J. Zhang, X. W. Liu, Supercloseness of linear finite element method on Bakhvalov-type meshes for singularly perturbed convection–diffusion equation in 1D, Appl. Math. Lett., 111 (2021), 106624. https://doi.org/10.1016/j.aml.2020.106624 doi: 10.1016/j.aml.2020.106624
[4]	X. Zhao, J. Zhang, Uniform convergence of finite element method on Vulanovi-Bakhvalov mesh for singularly perturbed convection–diffusion equation in 2D, Comput. Math. Appl., 188 (2025), 183–194. https://doi.org/10.1016/j.camwa.2025.04.007 doi: 10.1016/j.camwa.2025.04.007
[5]	Y. Wang, Y. H. Li, X. Y. Meng, An upwind finite volume element method on a Shishkin mesh for singularly perturbed convection–diffusion problems, J. Comput. Appl. Math., 438 (2024), 115493. https://doi.org/10.1016/j.cam.2023.115493 doi: 10.1016/j.cam.2023.115493
[6]	X. Y. Meng, M. Stynes, Energy-norm and balanced-norm supercloseness error analysis of a finite volume method on Shishkin meshes for singularly perturbed reaction–diffusion problems, Calcolo, 60 (2023), 40. https://doi.org/10.1007/s10092-023-00535-3 doi: 10.1007/s10092-023-00535-3
[7]	J. W. Zhou, Z. W. Jiang, H. T. Xie, H. F. Niu, The error estimates of spectral methods for 1-dimension singularly perturbed problem, Appl. Math. Lett., 100 (2020), 106001. https://doi.org/10.1016/j.aml.2019.106001 doi: 10.1016/j.aml.2019.106001
[8]	R. S. Temsah, Spectral methods for some singularly perturbed third order ordinary differential equations, Numer. Algor., 47 (2020), 63–80. https://doi.org/10.1007/s11075-007-9147-6 doi: 10.1007/s11075-007-9147-6
[9]	D. M. Tefera, A. A. Tiruneh, G. A. Derese, Fitted operator method using multiple fitting factors for two parameters singularly perturbed parabolic problems, Math. Probl. Eng., 2022 (2022), 6267522. https://doi.org/10.1155/2022/6267522 doi: 10.1155/2022/6267522
[10]	L. B. Liu, L. Xu, Y. Xu, Z. Huang, An adaptive grid method for a two-parameter singularly perturbed problem with non-smooth data, Appl. Math. Lett., 157 (2024), 109200. https://doi.org/10.1016/j.aml.2024.109200 doi: 10.1016/j.aml.2024.109200
[11]	A. G. R. Turnquist, Adaptive mesh methods on compact manifolds via optimal transport and optimal information transport, J. Comput. Phys., 500 (2024), 112726. https://doi.org/10.1016/j.jcp.2023.112726 doi: 10.1016/j.jcp.2023.112726
[12]	S. Franz, Convergence of local projection stabilisation finite element methods for convection–diffusion problems on layer-adapted meshes, Bit Numer. Math., 57 (2017), 771–786. https://doi.org/10.1007/s10543-017-0652-2 doi: 10.1007/s10543-017-0652-2
[13]	F. Erdogan, M. G. Saldn, O. Saldır, A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations, Appl. Math. Nonlinear Sci., 5 (2020), 425–436. http://dx.doi.org/10.2478/AMNS.2020.1.00040 doi: 10.2478/AMNS.2020.1.00040
[14]	N. S. Yadav, M. Kaushik, Stability and error analysis of an efficient numerical method for convection dominated parabolic PDEs with jump discontinuity in source function on modified layer-adapted mesh, Comput. Math. Math. Phys., 64 (2024), 509–536. https://doi.org/10.1134/S0965542524030102 doi: 10.1134/S0965542524030102
[15]	J. Mackenzie, Uniform convergence analysis of an upwind finite-difference approximation of a convection-diffusion boundary value problem on an adaptive grid, IMA J. Numer. Anal., 19 (1999), 233–249. https://doi.org/10.1093/imanum/19.2.233 doi: 10.1093/imanum/19.2.233
[16]	H. G. Roos, M. Stynes, L. Tobiska, Robust numerical methods for singularly perturbed differential equations, In: Springer series in computational mathematics, Berlin: Springer, 24 (2008). https://doi.org/10.1007/978-3-540-34467-4
[17]	T. Wang, T. Liu, A consistent fourth-order compact finite difference scheme for solving vorticity-stream function form of incompressible Navier-Stokes equations, Numer. Math. Theory Methods Appl., 12 (2019), 312–330. https://doi.org/10.4208/nmtma.OA-2018-0043 doi: 10.4208/nmtma.OA-2018-0043
[18]	S. K. Lete, T. G. Liu, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16–42. https://doi.org/10.1016/0021-9991(92)90324-R doi: 10.1016/0021-9991(92)90324-R
[19]	Z. F. Tian, Y. A. Li, Numerical solution of the incompressible Navier-Stokes equations with a three-point fourth-order upwind compact difference schemes, In: Proceedings of fourth international conference nonlinear mechanic, 2002,942–946.
[20]	J. C. Zhao, T. Zhang, R. M. Corless, Convergence of the compact finite difference method for second-order elliptic equations, Appl. Math. Comput., 182 (2006), 1451–1469. https://doi.org/10.1016/j.amc.2006.05.033 doi: 10.1016/j.amc.2006.05.033

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)