A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances

Fatemeh Sharafi

doi:doi:10.11648/j.acm.20261506.12

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A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances

Fatemeh Sharafi^*

Published in Applied and Computational Mathematics (Volume 15, Issue 6)

Received: 20 September 2026 Accepted: 4 October 2026 Published: 14 November 2026

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Abstract

This paper presents an advanced analysis of the radial basis function (RBF) collocation method for solving time-dependent partial differential equations (PDEs). A unified framework is developed that integrates weighted finite difference temporal discretization with global RBF spatial approximation for both smooth and piecewise smooth radial kernels. We provide a rigorous stability analysis of the resulting time-stepping scheme and systematically examine the roles of the shape parameter, polynomial augmentation, and scattered point distribution in controlling accuracy and conditioning. In addition to the classical global collocation formulation, we survey and compare contemporary localized and high-order RBF variants—including localized RBF collocation (LRBF), RBF partition of unity (RBF-PU), ghost-point-enhanced RBF pseudo-spectral (RBF-PS-G), and RBF finite difference (RBF-FD)—that have emerged as the state of the art for large-scale and long-time simulations in 2024–2026. Numerical experiments on benchmark problems including the telegraph and Klein–Gordon equations, a nonlinear viscous Burgers equation, a three-dimensional Helmholtz problem, and comparisons between global RBF, RBF-PU, and LRBF validate the theoretical predictions. The method's simplicity, mesh-free nature, and high-order accuracy make it a competitive and highly adaptable tool for a broad class of PDE problems.

Published in	Applied and Computational Mathematics (Volume 15, Issue 6)
DOI	10.11648/j.acm.20261506.12
Page(s)	233-243
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Meshless Methods, Radial Basis Functions, Collocation Method, Partial Differential Equations, Stability Analysis, Shape Parameter

References

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APA Style

Sharafi, F. (2026). A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances. Applied and Computational Mathematics, 15(6), 233-243. https://doi.org/10.11648/j.acm.20261506.12

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ACS Style

Sharafi, F. A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances. Appl. Comput. Math. 2026, 15(6), 233-243. doi: 10.11648/j.acm.20261506.12

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AMA Style

Sharafi F. A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances. Appl Comput Math. 2026;15(6):233-243. doi: 10.11648/j.acm.20261506.12

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@article{10.11648/j.acm.20261506.12,
  author = {Fatemeh Sharafi},
  title = {A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances},
  journal = {Applied and Computational Mathematics},
  volume = {15},
  number = {6},
  pages = {233-243},
  doi = {10.11648/j.acm.20261506.12},
  url = {https://doi.org/10.11648/j.acm.20261506.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261506.12},
  abstract = {This paper presents an advanced analysis of the radial basis function (RBF) collocation method for solving time-dependent partial differential equations (PDEs). A unified framework is developed that integrates weighted finite difference temporal discretization with global RBF spatial approximation for both smooth and piecewise smooth radial kernels. We provide a rigorous stability analysis of the resulting time-stepping scheme and systematically examine the roles of the shape parameter, polynomial augmentation, and scattered point distribution in controlling accuracy and conditioning. In addition to the classical global collocation formulation, we survey and compare contemporary localized and high-order RBF variants—including localized RBF collocation (LRBF), RBF partition of unity (RBF-PU), ghost-point-enhanced RBF pseudo-spectral (RBF-PS-G), and RBF finite difference (RBF-FD)—that have emerged as the state of the art for large-scale and long-time simulations in 2024–2026. Numerical experiments on benchmark problems including the telegraph and Klein–Gordon equations, a nonlinear viscous Burgers equation, a three-dimensional Helmholtz problem, and comparisons between global RBF, RBF-PU, and LRBF validate the theoretical predictions. The method's simplicity, mesh-free nature, and high-order accuracy make it a competitive and highly adaptable tool for a broad class of PDE problems.},
 year = {2026}
}

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TY  - JOUR
T1  - A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances
AU  - Fatemeh Sharafi
Y1  - 2026/11/14
PY  - 2026
N1  - https://doi.org/10.11648/j.acm.20261506.12
DO  - 10.11648/j.acm.20261506.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 233
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AB  - This paper presents an advanced analysis of the radial basis function (RBF) collocation method for solving time-dependent partial differential equations (PDEs). A unified framework is developed that integrates weighted finite difference temporal discretization with global RBF spatial approximation for both smooth and piecewise smooth radial kernels. We provide a rigorous stability analysis of the resulting time-stepping scheme and systematically examine the roles of the shape parameter, polynomial augmentation, and scattered point distribution in controlling accuracy and conditioning. In addition to the classical global collocation formulation, we survey and compare contemporary localized and high-order RBF variants—including localized RBF collocation (LRBF), RBF partition of unity (RBF-PU), ghost-point-enhanced RBF pseudo-spectral (RBF-PS-G), and RBF finite difference (RBF-FD)—that have emerged as the state of the art for large-scale and long-time simulations in 2024–2026. Numerical experiments on benchmark problems including the telegraph and Klein–Gordon equations, a nonlinear viscous Burgers equation, a three-dimensional Helmholtz problem, and comparisons between global RBF, RBF-PU, and LRBF validate the theoretical predictions. The method's simplicity, mesh-free nature, and high-order accuracy make it a competitive and highly adaptable tool for a broad class of PDE problems.
VL  - 15
IS  - 6
ER  -

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Author Information

Fatemeh Sharafi

Department of Mathematics, Islamic Azad University, Kerman, Iran

Contact Email

http://orcid.org/0000-0002-1825-0097

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Sharafi, F. (2026). A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances. Applied and Computational Mathematics, 15(6), 233-243. https://doi.org/10.11648/j.acm.20261506.12

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ACS Style

Sharafi, F. A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances. Appl. Comput. Math. 2026, 15(6), 233-243. doi: 10.11648/j.acm.20261506.12

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AMA Style

Sharafi F. A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances. Appl Comput Math. 2026;15(6):233-243. doi: 10.11648/j.acm.20261506.12

Copy | Download

@article{10.11648/j.acm.20261506.12,
  author = {Fatemeh Sharafi},
  title = {A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances},
  journal = {Applied and Computational Mathematics},
  volume = {15},
  number = {6},
  pages = {233-243},
  doi = {10.11648/j.acm.20261506.12},
  url = {https://doi.org/10.11648/j.acm.20261506.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261506.12},
  abstract = {This paper presents an advanced analysis of the radial basis function (RBF) collocation method for solving time-dependent partial differential equations (PDEs). A unified framework is developed that integrates weighted finite difference temporal discretization with global RBF spatial approximation for both smooth and piecewise smooth radial kernels. We provide a rigorous stability analysis of the resulting time-stepping scheme and systematically examine the roles of the shape parameter, polynomial augmentation, and scattered point distribution in controlling accuracy and conditioning. In addition to the classical global collocation formulation, we survey and compare contemporary localized and high-order RBF variants—including localized RBF collocation (LRBF), RBF partition of unity (RBF-PU), ghost-point-enhanced RBF pseudo-spectral (RBF-PS-G), and RBF finite difference (RBF-FD)—that have emerged as the state of the art for large-scale and long-time simulations in 2024–2026. Numerical experiments on benchmark problems including the telegraph and Klein–Gordon equations, a nonlinear viscous Burgers equation, a three-dimensional Helmholtz problem, and comparisons between global RBF, RBF-PU, and LRBF validate the theoretical predictions. The method's simplicity, mesh-free nature, and high-order accuracy make it a competitive and highly adaptable tool for a broad class of PDE problems.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - A Radial Basis Function Collocation Method for Time-Dependent Partial Differential Equations: Stability, Accuracy, and Recent Advances
AU  - Fatemeh Sharafi
Y1  - 2026/11/14
PY  - 2026
N1  - https://doi.org/10.11648/j.acm.20261506.12
DO  - 10.11648/j.acm.20261506.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 233
EP  - 243
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20261506.12
AB  - This paper presents an advanced analysis of the radial basis function (RBF) collocation method for solving time-dependent partial differential equations (PDEs). A unified framework is developed that integrates weighted finite difference temporal discretization with global RBF spatial approximation for both smooth and piecewise smooth radial kernels. We provide a rigorous stability analysis of the resulting time-stepping scheme and systematically examine the roles of the shape parameter, polynomial augmentation, and scattered point distribution in controlling accuracy and conditioning. In addition to the classical global collocation formulation, we survey and compare contemporary localized and high-order RBF variants—including localized RBF collocation (LRBF), RBF partition of unity (RBF-PU), ghost-point-enhanced RBF pseudo-spectral (RBF-PS-G), and RBF finite difference (RBF-FD)—that have emerged as the state of the art for large-scale and long-time simulations in 2024–2026. Numerical experiments on benchmark problems including the telegraph and Klein–Gordon equations, a nonlinear viscous Burgers equation, a three-dimensional Helmholtz problem, and comparisons between global RBF, RBF-PU, and LRBF validate the theoretical predictions. The method's simplicity, mesh-free nature, and high-order accuracy make it a competitive and highly adaptable tool for a broad class of PDE problems.
VL  - 15
IS  - 6
ER  -

Copy | Download