In this paper, an adaptive grid method is put forward to solve a singularly perturbed convection-diffusion problem with a discontinuous convection coefficient. First, this problem is discretized by using an upwind finite difference scheme on an arbitrary nonuniform grid except the fixed jump point. Then, a first-order maximum norm a posterior error estimate is derived. Further, based on this a posteriori error estimation and the mesh equidistribution principle, an adaptive grid generation algorithm is constructed. Finally, some numerical experiments are presented that support our theoretical estimate.
Citation: Xiongfa Mai, Ciwen Zhu, Libin Liu. An adaptive grid method for a singularly perturbed convection-diffusion equation with a discontinuous convection coefficient[J]. Networks and Heterogeneous Media, 2023, 18(4): 1528-1538. doi: 10.3934/nhm.2023067
In this paper, an adaptive grid method is put forward to solve a singularly perturbed convection-diffusion problem with a discontinuous convection coefficient. First, this problem is discretized by using an upwind finite difference scheme on an arbitrary nonuniform grid except the fixed jump point. Then, a first-order maximum norm a posterior error estimate is derived. Further, based on this a posteriori error estimation and the mesh equidistribution principle, an adaptive grid generation algorithm is constructed. Finally, some numerical experiments are presented that support our theoretical estimate.
| [1] | K. W. Chang, F. A. Howes, Nonlinear Singular Perturbation Phenomena: Theory and Application, New York: Springer, 1984. |
| [2] | H. G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equation, Berlin: Springer, 2008. |
| [3] |
P. A. Farrell, A. F. Hegarty, J. Miller, Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient, Math. Comput. Model., 40 (2004), 1375–1392. https://doi.org/10.1016/j.mcm.2005.01.025 doi: 10.1016/j.mcm.2005.01.025
|
| [4] |
Z. Cen, A hybrid difference scheme for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient, Appl. Math. Comput., 169 (2005), 689–699. https://doi.org/10.1016/j.amc.2004.08.051 doi: 10.1016/j.amc.2004.08.051
|
| [5] | V. Shanthi, N. Ramanujam, S. Natesan, Fitted mesh method for singularly perturbed reaction-convection-diffusion problems with boundary and interior layers, J. Appl. Math. Comput., 22 (2006), 49–65. |
| [6] | M. Chandru, T. Prabha, V. Shanthi, A parameter robust higher order numerical method for singularly perturbed two parameter problems with non-smooth data, J. Comput. Appl. Math., 309 (2017), 11–27. |
| [7] | T. Prabha, M. Chandru, V. Shanthi, Hybrid difference scheme for singularly perturbed reaction-convection-diffusion problem with boundary and interior layers, Appl. Math. Comput., 314 (2017), 237–256. |
| [8] |
R. M. Priyadharshini, N. Ramanujam, Approximation of derivative to a singularly perturbed second-order ordinary differential equation with discontinuous convection coefficient using hybrid difference scheme, Int. J. Comput. Math., 86 (2009), 1355–1365. https://doi.org/10.1080/00207160701870837 doi: 10.1080/00207160701870837
|
| [9] |
A. Tamilselvan, N. Ramanujam, A parameter uniform numerical method for a system of singularly perturbed convection-diffusion equations with discontinuous convection coefficients, Int. J. Comput. Math., 87 (2010), 1374–1388. https://doi.org/10.1080/00207160802322332 doi: 10.1080/00207160802322332
|
| [10] |
M. B. Pathan, S. Vembu, A parameter-uniform second order numerical method for a weakly coupled system of singularly perturbed convection-diffusion equations with discontinuous convection coefficients and source terms, Calcolo, 54 (2017), 1027–1053. https://doi.org/10.1007/s10092-017-0218-3 doi: 10.1007/s10092-017-0218-3
|
| [11] | K. Aarthika, R. Shiromani, V. Shanthi, A higher-order finite difference method for two-dimensional singularly perturbed reaction-diffusion with source-term-discontinuous problem, Comput. Math. Appl., 118 (2022), 56–73. |
| [12] |
K. Mukherjee, S. Natesan, Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficients, BIT Numer. Math., 51 (2011), 289–315. https://doi.org/10.1111/j.1475-6765.2011.02006.x doi: 10.1111/j.1475-6765.2011.02006.x
|
| [13] |
S. C. S. Rao, S. Chawla, A. K. Chaturvedi, Numerical analysis for a class of coupled system of singularly perturbed time-dependent convection-diffusion equations with a discontinuous source term, Numer Methods Partial Differ Equ, 38 (2022), 1437–1467. https://doi.org/10.1002/num.22845 doi: 10.1002/num.22845
|
| [14] | M. Chandru, P. Das, H. Ramos, Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data, Math. Methods Appl. Sci., 41 (2018), 5359–5387. |
| [15] | C. Clavero, J L. Gracia, G I. Shishkin, An efficient numerical scheme for 1D parabolic singularly perturbed problems with an interior and boundary layers, J. Comput. Appl. Math., 318 (2015), 634–645. |
| [16] |
A. Kaushik, N. Sharma, An adaptive difference scheme for parabolic delay differential equation with discontinuous coefficients and interior layers, J. Difference Equ. Appl., 26 (2020), 1450–1470. https://doi.org/10.1080/10236198.2020.1843645 doi: 10.1080/10236198.2020.1843645
|
| [17] | S. C. S. Rao, S. Chawla, Numerical solution of singularly perturbed linear parabolic system with discontinuous source term, Appl. Numer. Math., 127 (2008), 249–265. |
| [18] |
K. Aarthika, V. Shanthi, H. Ramos, A computational approach for a two-parameter singularly perturbed system of partial differential equations with discontinuous coefficients, Appl. Math. Comput., 434 (2022), 127409. https://doi.org/10.1016/j.amc.2022.127409 doi: 10.1016/j.amc.2022.127409
|
| [19] |
S. C. S. Rao, A. K. Chaturvedi, Analysis and implementation of a computational technique for a coupled system of two singularly perturbed parabolic semilinear reaction-diffusion equations having discontinuous source terms, Commun. Nonlinear Sci. Numer. Simul., 108 (2022), 106232. https://doi.org/10.1016/j.cnsns.2021.106232 doi: 10.1016/j.cnsns.2021.106232
|
| [20] |
N. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), 423–441. https://doi.org/10.1137/S0036142900368642 doi: 10.1137/S0036142900368642
|
| [21] |
N. Kopteva, M. Stynes, A robust adaptive method for quasi-linear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), 1446–1467. https://doi.org/10.1137/S003614290138471X doi: 10.1137/S003614290138471X
|
| [22] |
Y. Chen, Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid, Adv. Comput. Math., 24 (2006), 197–212. https://doi.org/10.1007/s10444-004-7641-0 doi: 10.1007/s10444-004-7641-0
|
| [23] |
L. B. Liu, Y. Chen, Maximum norm a posteriori error estimates for a singularly perturbed differential difference equation with small delay, Appl. Math. Comput., 227 (2014), 801–810. https://doi.org/10.1016/j.amc.2013.10.085 doi: 10.1016/j.amc.2013.10.085
|
| [24] |
T. Linß, Analysis of a system of singularly perturbed convection-diffusion equations with strong coupling, SIAM J. Numer. Anal., 47 (2009), 1847–1862. https://doi.org/10.1137/070683970 doi: 10.1137/070683970
|
| [25] |
L. B. Liu, Y. Chen, A robust adaptive grid method for a system of two singularly perturbed convection-diffusion equations with weak coupling, J. Sci. Comput., 61 (2014), 1–16. https://doi.org/10.1007/s10915-013-9814-9 doi: 10.1007/s10915-013-9814-9
|
| [26] |
L. B. Liu, Y. Chen, A-posteriori error estimation in maximum norm for a strongly coupled system of two singularly perturbed convection-diffusion problems, J. Comput. Appl. Math., 313 (2017), 152–167. https://doi.org/10.1016/j.cam.2016.08.020 doi: 10.1016/j.cam.2016.08.020
|
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Xiongfa Mai, Ciwen Zhu, Libin Liu. An adaptive grid method for a singularly perturbed convection-diffusion equation with a discontinuous convection coefficient[J]. Networks and Heterogeneous Media, 2023, 18(4): 1528-1538. doi: 10.3934/nhm.2023067
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