The stability and convergence of time-stepping/spectral methods with asymptotic behaviour for the Rosenau–Burgers equation

ABSTRACT

This study is concerned with numerical methods for the periodic initial boundary value problem of Rosenau–Burgers equation. The main objective is to construct and analyze stable time stepping scheme for the time discretization and use a spectral method for the spatial discretization. It is proved that the combined method is unconditionally stable, and convergent in order of $\mathcal{O}(\mathrm{\Delta }{t}^2+N^{1-m})$, where $\mathrm{\Delta }t$, N and m are respectively time step size, polynomial degree, and regularity in the space variable. The numerical tests carried out in the paper show that the proposed method is efficient. In addition, the asymptotic analysis of the solution is discussed by using some existing theorem.

COMMUNICATED BY:

• Y. C. Hon

Keywords:

• Rosenau–Burgers
• finite difference/spectral approximations
• stability
• convergence

2010 Mathematics Subject Classification:

• 65Mxx

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This Research is Partially Supported by the National Natural Science Foundation of China [grant numbers 11471274, 11421110001, 51661135011, and 91630204].