Adaptive Multiquadratic Radial Basis Function-based Explicit Runge--Kutta Methods
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911041525383168 |
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| author | Yadav, Rajesh Yadav, Deepak Kumar Kumar, Alpesh |
| author_facet | Yadav, Rajesh Yadav, Deepak Kumar Kumar, Alpesh |
| contents | Runge--Kutta (RK) methods are widely used techniques for solving a class of initial value problems. In this article, we introduce an adaptive multiquadratic (MQ) radial basis function (RBF)-based method to develop enhanced explicit RK methods. These methods achieve a higher order of convergence than the corresponding classical RK methods. To improve the local convergence of the numerical solution, we optimize the free parameters (shape functions) involved in the RBFs by forcing the local truncation errors to vanish. We also present a convergence and stability analysis of the proposed methods. To demonstrate the advantages of these methods in terms of accuracy and convergence, we consider several numerical examples and compare the performance of our methods with that of the classical RK methods. The Tables and Figures presented in this article clearly validate the superiority of the proposed methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04261 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive Multiquadratic Radial Basis Function-based Explicit Runge--Kutta Methods Yadav, Rajesh Yadav, Deepak Kumar Kumar, Alpesh Numerical Analysis 65L06, 65L05, 65D12 Runge--Kutta (RK) methods are widely used techniques for solving a class of initial value problems. In this article, we introduce an adaptive multiquadratic (MQ) radial basis function (RBF)-based method to develop enhanced explicit RK methods. These methods achieve a higher order of convergence than the corresponding classical RK methods. To improve the local convergence of the numerical solution, we optimize the free parameters (shape functions) involved in the RBFs by forcing the local truncation errors to vanish. We also present a convergence and stability analysis of the proposed methods. To demonstrate the advantages of these methods in terms of accuracy and convergence, we consider several numerical examples and compare the performance of our methods with that of the classical RK methods. The Tables and Figures presented in this article clearly validate the superiority of the proposed methods. |
| title | Adaptive Multiquadratic Radial Basis Function-based Explicit Runge--Kutta Methods |
| topic | Numerical Analysis 65L06, 65L05, 65D12 |
| url | https://arxiv.org/abs/2507.04261 |