Stability of θ-schemes for partial differential equations with piecewise constant arguments of alternately retarded and advanced type

ABSTRACT

This paper deals with the analytical and numerical stability of a partial differential equation with piecewise constant arguments of alternately retarded and advanced type. Firstly, the theory of separation of variables in matrix form and the Fourier method are implemented to achieve the sufficient conditions under which the analytic solution is asymptotically stable. Secondly, the discrete equation is obtained by applying the θ-schemes to the original continuous equation, the sufficient conditions for the asymptotic stability of numerical solution are also shown when the mesh ratio satisfying certain conditions. Finally, some numerical experiments for verifying the theoretical results are provided.

KEYWORDS:

• Partial differential equation
• piecewise constant arguments
• θ-schemes
• analytical stability
• numerical stability

2010 AMS SUBJECT CLASSIFICATIONS:

• 35B35
• 65M06
• 65M12

Acknowledgments

The authors are grateful to the editor and the anonymous referees for their helpful comments and suggestions on the revision of the manuscript and they also wish to thank Professors Mingzhu Liu and Hui Liang for their selfless help to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under grant [11201084] and the Natural Science Foundation of Guangdong Province under grant [2017A030313031].