Streamlined numerical solutions of burgers' equations bridging the tailored finite point method and the Cole Hopf transformation

Abstract

This paper explores the possibility of solving the one-dimensional Burgers' equation with a tailored finite point method (TFPM) based on an explicit stencil along with utilizing the Cole Hopf transformation. The proposed TFPM procedure operates on an explicit four-point stencil where the nodal solution at the advanced temporal level is written as a linear combination of the nodal solutions at the preceding temporal level. In order to bring the essence of the local exact solutions into the numerical approximations the scalar coefficients involved in the linear sum are determined by the application of the fundamental solutions into the stencil. The efficiency of the method is demonstrated through the comparisons of TFPM solutions for various examples with the exact solutions and solutions from well-established methodologies in the literature.

Keywords:

• Burgers' equation
• TFPM
• 4-point centred stencil
• stability
• convergence

2020 Mathematics Subject Classification:

• 65N12

Acknowledgments

The authors would like to convey their gratitude to the anonymous reviewers for their meaningful observations, positive critiques, and beneficial recommendations. The development of the proposed algorithm has been a collective endeavour involving all three authors, with a focus on conceptualization, formulation, and analysis. V P Shyaman and A Sreelakshmi assumed responsibility for coding, data tabulation, and experiment documentation under the close mentorship of Ashish Awasthi. Furthermore, the drafting of the manuscript was a collaborative effort between V P Shyaman and A Sreelakshmi, with continual guidance and input from Ashish Awasthi throughout the entire process.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Availability of supporting data

The present study did not involve the generation or analysis of any data sets.

Additional information

Funding

Financial support, in the form of research grants, was extended to the authors by the Ministry of Human Resource Development, Government of India, and the University Grants Commission, Government of India.