A multi-domain Galerkin method with numerical integration for the Burgers equation

Abstract

A multi-domain Galerkin method is presented for the Burgers equation, whose solutions behave differently at positive infinity and negative infinity. The original problem is transformed into a problem with general variable coefficients and homogeneous boundary conditions at infinities by using the auxiliary function, which is capable of being reformulated as a suitable variational formulation that is the most convenient for analysing numerical error. As an application, the composite spectral scheme with numerical integration is provided for the underlying problem on the whole line, which is easy to deal with problems concerning general variable coefficients. Some composite generalized Laguerre–Legendre interpolations on the whole line are established. The convergence and stability of the proposed algorithm are proved. Numerical results show the efficiency of this approach and agree well with the theoretical analysis.

Keywords:

• Multi-domain Galerkin method with numerical integration
• composite generalized Laguerre–Legendre interpolation approximations
• Burgers equation
• the Cauchy problem
• the whole line

2020 AMS SUBJECT CLASSIFICATIONS:

• 65M70
• 41A30
• 35K55

Acknowledgments

The authors would like to thank the anonymous referees, Dr Yue Zhu of Hangzhou Dianzi University, Dr Lu-yu Wang of Zhejiang University and Jun-jie Zheng of East China University of Political Science and Law for their help on polishing this work.

Disclosure statement

The author confirm that this article has no competing interests with any institution or individual.

Additional information

Funding

The author of this work is supported in part by NSF of China (grant numbers 12171141, 12271365, 11371123) and NSF of Henan Province of China [grant number 202300410156].