High-order numerical solution of viscous Burgers' equation using an extended Cole–Hopf barycentric Gegenbauer integral pseudospectral method

Abstract

This paper presents an extension to the Cole–Hopf barycentric Gegenbauer integral pseudospectral (PS) method (CHBGPM) presented in Elgindy and Dahy [High-order numerical solution of viscous Burgers' equation using a Cole–Hopf barycentric Gegenbauer integral pseudospectral method, Math. Methods Appl. Sci. 41 (2018), pp. 6226–6251] to solve an initial-boundary value problem of Burgers' type when the boundary function k defined at the right boundary of the spatial domain vanishes at a finite set of real numbers or on a single/multiple subdomain(s) of the solution domain. We present a new strategy that is computationally more efficient than that presented in [12] in the former case, and can be implemented successfully in the latter case when the method of [12] fails to work. Moreover, fully exponential convergence rates are still preserved in both spatial and temporal directions if the boundary function k is sufficiently smooth. Numerical comparisons with other traditional methods in the literature are presented to confirm the efficiency of the proposed method. A numerical study of the condition number of the linear systems produced by the method is included.

Keywords:

• Barycentric
• Burgers' equation
• Cole–Hopf
• Gegenbauer polynomials
• Gegenbauer quadrature
• pseudospectral

2010 Mathematics Subject Classifications:

• Numerical analysis
• 90-XX operations research
• mathematical programming
• partial differential equations

Disclosure statement

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