Numerical efficiency of some exponential methods for an advection–diffusion equation

ABSTRACT

In this paper, we investigate several modified exponential finite-difference methods to approximate the solution of the one-dimensional viscous Burgers' equation. Burgers' equation admits solutions that are positive and bounded under appropriate conditions. Motivated by these facts, we propose nonsingular exponential methods that are capable of preserving some structural properties of the solutions of Burgers' equation. The fact that some of the techniques preserve structural properties of the solutions is thoroughly established in this work. Rigorous analyses of consistency, stability and numerical convergence of these schemes are presented for the first time in the literature, together with estimates of the numerical solutions. The methods are computationally improved for efficiency using the Padé approximation technique. As a result, the computational cost is substantially reduced in this way. Comparisons of the numerical approximations against the exact solutions of some initial-boundary-value problems for different Reynolds numbers show a good agreement between them.

KEYWORDS:

• Viscous Burgers' equation
• modified exponential finite-difference scheme
• Padé approximation techniques
• structure-preserving method
• stability and convergence analyses

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 35K15

Acknowledgments

The authors wish to thank the anonymous reviewers and the editor in charge of handling this manuscript for all their invaluable comments. Their constructive criticisms and suggestions enabled us to improve the quality of this work. Special thanks go to Reviewer #1, who pointed out a mistake in the proof of Theorem 5.4.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Jorge Eduardo Macías-Dvaz http://orcid.org/0000-0002-7580-7533

Bilge İnanb http://orcid.org/0000-0001-5166-0146