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Uniform convergence analysis of a new adaptive upwind finite difference method for singularly perturbed convection-reaction-diffusion boundary value problems

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Abstract

In this paper, a new adaptive upwind finite difference method based on the arc-length equidistribution principle is studied for solving the general linear singularly perturbed convection-reaction-diffusion two-point boundary value problem. Under the discrete comparison principle and its derived properties, the a posteriori error estimate of the upwind finite difference scheme on an arbitrary mesh is obtained by using a linear interpolation, the existence of the solution of the adaptive upwind finite difference method is proved without the presupposition, and the boundedness of the arc-length of the numerical solution and then the uniform first-order convergence of the adaptive upwind finite difference method are achieved by the discrete Green’s function. Finally, the proposed method is verified to be uniformly first-order convergent and is compared with the existing method in numerical examples

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Acknowledgements

The authors thank the anonymous reviewers for their valuable comments which help to improve the presentation. This paper is supported by National Natural Science Foundation (Grant No. 11471019) of China.

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Authors and Affiliations

  1. College of Sciences, North China University of Technology, Bei**g, 100144, China

    Quan Zheng

  2. Institute of Mathematics and Physics, Bei**g Union University, Bei**g, 100101, China

    Zhongli Liu

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  1. Quan Zheng
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  2. Zhongli Liu
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Correspondence to Quan Zheng.

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Zheng, Q., Liu, Z. Uniform convergence analysis of a new adaptive upwind finite difference method for singularly perturbed convection-reaction-diffusion boundary value problems. J. Appl. Math. Comput. 70, 601–618 (2024). https://doi.org/10.1007/s12190-023-01977-2

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  • DOI: https://doi.org/10.1007/s12190-023-01977-2

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