Computational study of the convection-diffusion equation using new cubic B-spline approximations

Asifa Tassaddiq; Muhammad Yaseen; Aatika Yousaf; Rekha Srivastava; Asifa Tassaddiq; Muhammad Yaseen; Aatika Yousaf; Rekha Srivastava

doi:10.3934/math.2021259

AIMS Mathematics

2021, Volume 6, Issue 5: 4370-4393. doi: 10.3934/math.2021259

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Research article

Computational study of the convection-diffusion equation using new cubic B-spline approximations

1.
Department of Basic Sciences and Humanities, College of Computer and Information Sciences Majmaah University, Al-Majmaah 11952, Saudi Arabia
2.
Department of Mathematics, University of Sargodha, 40100, Pakistan
3.
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada

Received: 11 December 2020 Accepted: 08 February 2021 Published: 19 February 2021
MSC : 65M70, 65Z05, 65D05, 65D07, 35B35

This paper introduces an efficient numerical procedure based on cubic B-Spline (CuBS) with a new approximation for the second-order space derivative for computational treatment of the convection-diffusion equation (CDE). The time derivative is approximated using typical finite differences. The key benefit of the scheme is that the numerical solution is obtained as a smooth piecewise continuous function which empowers one to find approximate solution at any desired position in the domain. Further, the new approximation has considerably increased the accuracy of the scheme. A stability analysis is performed to assure that the errors do not magnify. Convergence analysis of the scheme is also discussed. The scheme is implemented on some test problems and the outcomes are contrasted with those of some current approximating techniques from the literature. It is concluded that the offered scheme is equitably superior and effective.
- convection-diffusion equation,
- cubic B-splines,
- stability,
- new cubic B-spline approximations
Citation: Asifa Tassaddiq, Muhammad Yaseen, Aatika Yousaf, Rekha Srivastava. Computational study of the convection-diffusion equation using new cubic B-spline approximations[J]. AIMS Mathematics, 2021, 6(5): 4370-4393. doi: 10.3934/math.2021259

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Abstract

This paper introduces an efficient numerical procedure based on cubic B-Spline (CuBS) with a new approximation for the second-order space derivative for computational treatment of the convection-diffusion equation (CDE). The time derivative is approximated using typical finite differences. The key benefit of the scheme is that the numerical solution is obtained as a smooth piecewise continuous function which empowers one to find approximate solution at any desired position in the domain. Further, the new approximation has considerably increased the accuracy of the scheme. A stability analysis is performed to assure that the errors do not magnify. Convergence analysis of the scheme is also discussed. The scheme is implemented on some test problems and the outcomes are contrasted with those of some current approximating techniques from the literature. It is concluded that the offered scheme is equitably superior and effective.

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