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Integrated radial basis functions (IRBFs) to simulate nonlinear advection–diffusion equations with smooth and non-smooth initial data

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Abstract

In this article, a meshfree method for the numerical solution of conversation law equations is considered. Some problems which have shock such as advection problems are not properly solved by radial basis function collocation meshfree method. Therefore, we use the integrated radial basis function (IRBF) method for some of these problems. In the current study, the governing models have been discretized by IRBF technique in the spatial direction and by finite difference approximation for time variable. This converts the main problem to a system of nonlinear ordinary differential equations (ODEs). Furthermore, the obtained ODEs will be solved by Runge–Kutta technique. This is the meshless method of lines technique. Numerical examples indicate the acceptable accuracy, proficiency and easy implementation of the presented method.

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Acknowledgements

We would like to appreciate reviewers that their comments and suggestions improved this paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Ali Ebrahimijahan, Mehdi Dehghan & Mostafa Abbaszadeh

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  1. Ali Ebrahimijahan
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  2. Mehdi Dehghan
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  3. Mostafa Abbaszadeh
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Correspondence to Mehdi Dehghan.

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Ebrahimijahan, A., Dehghan, M. & Abbaszadeh, M. Integrated radial basis functions (IRBFs) to simulate nonlinear advection–diffusion equations with smooth and non-smooth initial data. Engineering with Computers 38, 1071–1106 (2022). https://doi.org/10.1007/s00366-020-01039-2

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