An Integral-like Numerical Approach for Solving Burgers’ Equation

Somrath Kanoksirirath

doi:doi:10.11648/j.pamj.20241302.11

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An Integral-like Numerical Approach for Solving Burgers’ Equation

Somrath Kanoksirirath^*

Published in Pure and Applied Mathematics Journal (Volume 13, Issue 2)

Received: 24 April 2024 Accepted: 15 May 2024 Published: 12 June 2024

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Abstract

The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the ӏ₁, ӏ₂, and ӏ_∞ error norms down to the order of 10⁻⁴. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is ν∆t/∆x² > 0.02, including the viscosity coefficient ν due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆t even when using a small grid spacing ∆x, emphasizing its suitability for practical applications such as numerical weather prediction.

Published in	Pure and Applied Mathematics Journal (Volume 13, Issue 2)
DOI	10.11648/j.pamj.20241302.11
Page(s)	17-28
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), 2024. Published by Science Publishing Group

Keywords

Burgers’ Equation, Hopf-Cole Transformation, Explicit Scheme, Parallel Scalability

References

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Kanoksirirath, S. (2024). An Integral-like Numerical Approach for Solving Burgers’ Equation. Pure and Applied Mathematics Journal, 13(2), 17-28. https://doi.org/10.11648/j.pamj.20241302.11

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ACS Style

Kanoksirirath, S. An Integral-like Numerical Approach for Solving Burgers’ Equation. Pure Appl. Math. J. 2024, 13(2), 17-28. doi: 10.11648/j.pamj.20241302.11

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Kanoksirirath S. An Integral-like Numerical Approach for Solving Burgers’ Equation. Pure Appl Math J. 2024;13(2):17-28. doi: 10.11648/j.pamj.20241302.11

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@article{10.11648/j.pamj.20241302.11,
  author = {Somrath Kanoksirirath},
  title = {An Integral-like Numerical Approach for Solving Burgers’ Equation},
  journal = {Pure and Applied Mathematics Journal},
  volume = {13},
  number = {2},
  pages = {17-28},
  doi = {10.11648/j.pamj.20241302.11},
  url = {https://doi.org/10.11648/j.pamj.20241302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241302.11},
  abstract = {The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the ӏ1, ӏ2, and ӏ∞ error norms down to the order of 10−4. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is ν∆t/∆x2 > 0.02, including the viscosity coefficient ν due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆t even when using a small grid spacing ∆x, emphasizing its suitability for practical applications such as numerical weather prediction.},
 year = {2024}
}

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TY  - JOUR
T1  - An Integral-like Numerical Approach for Solving Burgers’ Equation
AU  - Somrath Kanoksirirath
Y1  - 2024/06/12
PY  - 2024
N1  - https://doi.org/10.11648/j.pamj.20241302.11
DO  - 10.11648/j.pamj.20241302.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 17
EP  - 28
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20241302.11
AB  - The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the ӏ1, ӏ2, and ӏ∞ error norms down to the order of 10−4. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is ν∆t/∆x2 > 0.02, including the viscosity coefficient ν due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆t even when using a small grid spacing ∆x, emphasizing its suitability for practical applications such as numerical weather prediction.
VL  - 13
IS  - 2
ER  -

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Author Information

Somrath Kanoksirirath

National Electronics and Computer Technology Center (NECTEC), National Science and Technology Development Agency (NSTDA), Pathum Thani, Thailand

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http://orcid.org/0000-0002-1595-120X

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APA Style

Kanoksirirath, S. (2024). An Integral-like Numerical Approach for Solving Burgers’ Equation. Pure and Applied Mathematics Journal, 13(2), 17-28. https://doi.org/10.11648/j.pamj.20241302.11

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ACS Style

Kanoksirirath, S. An Integral-like Numerical Approach for Solving Burgers’ Equation. Pure Appl. Math. J. 2024, 13(2), 17-28. doi: 10.11648/j.pamj.20241302.11

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AMA Style

Kanoksirirath S. An Integral-like Numerical Approach for Solving Burgers’ Equation. Pure Appl Math J. 2024;13(2):17-28. doi: 10.11648/j.pamj.20241302.11

Copy | Download

@article{10.11648/j.pamj.20241302.11,
  author = {Somrath Kanoksirirath},
  title = {An Integral-like Numerical Approach for Solving Burgers’ Equation},
  journal = {Pure and Applied Mathematics Journal},
  volume = {13},
  number = {2},
  pages = {17-28},
  doi = {10.11648/j.pamj.20241302.11},
  url = {https://doi.org/10.11648/j.pamj.20241302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241302.11},
  abstract = {The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the ӏ1, ӏ2, and ӏ∞ error norms down to the order of 10−4. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is ν∆t/∆x2 > 0.02, including the viscosity coefficient ν due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆t even when using a small grid spacing ∆x, emphasizing its suitability for practical applications such as numerical weather prediction.},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - An Integral-like Numerical Approach for Solving Burgers’ Equation
AU  - Somrath Kanoksirirath
Y1  - 2024/06/12
PY  - 2024
N1  - https://doi.org/10.11648/j.pamj.20241302.11
DO  - 10.11648/j.pamj.20241302.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 17
EP  - 28
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20241302.11
AB  - The Burgers’ equation, commonly appeared in the study of turbulence, fluid dynamics, shock waves, and continuum mechanics, is a crucial part of the dynamical core of any numerical weather model, influencing simulated meteorological phenomena. While past studies have suggested several robust numerical approaches for solving the equation, many are too complicated for practical adaptation and too computationally expensive for operational deployment. This paper introduces an unconventional approach based on spline polynomial interpolations and the Hopf-Cole transformation. Using Taylor expansion to approximate the exponential term in the Hopf-Cole transformation, the analytical solution of the simplified equation is discretized to form our proposed scheme. The scheme is explicit and adaptable for parallel computing, although certain types of boundary conditions need to be employed implicitly. Three distinct test cases were utilized to evaluate its accuracy, parallel scalability, and numerical stability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation perform equally well, managing to reduce the ӏ1, ӏ2, and ӏ∞ error norms down to the order of 10−4. Parallel scalability observed in the weak-scaling experiment depends on time step size but is generally as good as any explicit scheme. The stability condition is ν∆t/∆x2 > 0.02, including the viscosity coefficient ν due to the Hopf-Cole transformation step. From the stability condition, the schemes can run at a large time step size ∆t even when using a small grid spacing ∆x, emphasizing its suitability for practical applications such as numerical weather prediction.
VL  - 13
IS  - 2
ER  -

Copy | Download