The issue of numerical differentiation of functions with large gradients is considered. It is assumed that there is a decomposition of a given function of one variable into the sum of a regular component and a boundary layer component; the latter is responsible for the large gradients of the function and is known up to a factor. This decomposition is valid, in particular, for a solution of a singularly perturbed boundary value problem. However, the application of the classical polynomial formulas of numerical differentiation to functions with large gradients may produce significant errors. Numerical differentiation formulas that are exact on the boundary layer component are studied, and their error is estimated. Such formulas are proved to be more exact than the classical ones in the case of the presence of a boundary layer component. An approach to estimating the error of the proposed formulas is suggested, and its applicability is shown in particular cases. The results of numerical experiments are presented. These results comply with the obtained error estimates and show the advantage in accuracy of the proposed formulas.
Keywords: function of one variable, large gradients, boundary layer component, nonpolynomial formula for numerical differentiation, error estimation
Received April 4, 2024
Revised May 10, 2024
Accepted May 13, 2024
Funding Agency: The work was supported under state contract IM SB RAS no. FWNF-2022-0016.
Aleksander Ivanovich Zadorin. Dr. Phys.-Math. Sci., Prof., Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia, e-mail: zadorin@ofim.oscsbras.ru
REFERENCES
1. Bakhvalov N.S., Zhidkov N.P., Kobel’kov G.M. Chislennyye metody [Numerical Methods]. Moscow, Nauka Publ., 1987, 598 p.
2. Zadorin A.I., Zadorin N.A. Interpolation formula for functions with a boundary layer component and its application to derivatives calculation. Sib. Electron. Math. Rep., 2012, vol. 9, pp. 445–455.
3. Il’in A.M. Difference scheme for a differential equation with a small parameter affecting the highest derivative. Math. Notes, 1969, vol. 6, iss. 2, pp. 596–602. doi: 10.1007/BF01093706
4. Kellogg R.B., Tsan A. Analysis of some difference approximations for a singular perturbation problems without turning points. Math. Comput., 1978, vol. 32, pp. 1025–1039. doi: 10.1090/S0025-5718-1978-0483484-9
5. Zadorin A.I., Zadorin N.A. Non-polynomial interpolation of functions with large gradients and its application. Comput. Math. Math. Phys., 2021, vol. 61, no. 2, pp. 167–176. doi: 10.1134/S0965542521020147
6. Zadorin A.I. Formulas for numerical differentiation of functions with large gradients. Numer. Anal. Appl., 2023, vol. 16, no. 1, pp. 14–21. doi: 10.1134/S1995423923010020
7. Zadorin A.I. Analysis of numerical differentiation formulas in a boundary layer on a Shishkin grid. Numer. Anal. Appl., 2018, vol. 11, no. 3, pp. 193–203. doi: 10.1134/S1995423918030011
8. Shishkin G.I. Setochnyye approksimatsii singulyarno vozmushchennykh ellipticheskikh i parabolicheskikh uravneniy [Grid approximations of singularly perturbed elliptic and parabolic equations]. Yekaterinburg, UrO RAN Publ., 1992, 233 p. ISBN: 5-7691-0159-8 .
9. Zadorin A.I. Analysis of numerical differential formulas on a Bakhvalov mesh in the presence of a boundary layer. Comput. Math. Math. Phys., 2023, vol. 63, no. 2, pp. 175–183. doi: 10.1134/S0965542523020148
10. Bakhvalov N.S. The optimization of methods of solving boundary value problems with a boundary layer. USSR Comput. Math. Math. Phys., 1969, vol. 9, no. 4, pp. 139–166. doi: 10.1016/0041-5553(69)90038-X
11. Roos H.G. Layer-adapted meshes: milestones in 50 years of history. Preprint arXiv:1909.08273 , 2019, 16 p. Available at: https://arxiv.org/abs/1909.08273 . doi: 10.48550/arXiv.1909.08273
12. Vulanovic R. On numerical solution of a power layer problem. In: Proc. III Conf. “Numerical methods and approximation theory”, ed. G.V. Milovanovoć, University of Niš, 1988, pp. 423–431.
13. Dautov R.Z., Timerbaev M.R. Chislennyye metody. Priblizheniye funktsiy: uchebnoye posobiye [Numerical methods. Function approximation: tutorial]. Kazan, Kazan Univ. Publ., 2021, 123 p.
14. Kopteva N.V., Stynes M. Approximation of derivatives in a convection-diffusion two-point boundary value problem. Appl. Numer. Math., 2001, vol. 39, pp. 47–60. doi: 10.1016/S0168-9274(01)00051-4
15. Shishkin G.I. Approximations of solutions and derivatives for a singularly perturbed elliptic convection-diffusion equations. Math. Proc. Royal Irish Acad., 2003, vol. 103A, no. 4, pp. 169–201. doi: 10.1353/mpr.2003.0010
Cite this article as: A.I. Zadorin. Analysis of numerical differentiation formulas on a uniform grid in the presence of a boundary layer. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 106–116.
