A high-order implicit–explicit Runge–Kutta type scheme for the numerical solution of the Kuramoto–Sivashinsky equation

Abstract

This manuscript is concerned with the development and the implementation of a numerical scheme to study the spatio-temporal solution profile of the well-known Kuramoto–Sivashinsky equation with appropriate initial and boundary conditions. A fourth-order Runge–Kutta based implicit–explicit scheme in time along with compact higher-order finite difference scheme in space is introduced. The proposed scheme takes full advantage of the method of line (MOL) and partial fraction decomposition techniques, therefore, it just needs to solve two backward Euler-type linear systems at each time step to get the solution. Performance of the scheme is investigated by testing it on some test examples and by comparing numerical results with relevant known schemes. The numerical results showed that the proposed scheme is more accurate and reliable than existing schemes to solve Kuramoto–Sivashinsky equation.

Keywords:

• Implicit–explicit scheme
• Kuramoto–Sivashinsky equation
• partial fraction splitting technique
• compact finite difference
• exponential time differencing

2010 Mathematics Subject Classification:

• 65

Acknowledgments

The authors are grateful to anonymous referees for their valuable comments and suggestions which improved the quality of this manuscript. A. Chowdhury acknowledges support from the US National Science Foundation under grant number 1800798.

Disclosure statement

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

Additional information

Funding

A. Chowdhury acknowledges support from the US National Science Foundation [grant number 1800798].