A backward approach to certain class of transport equations in any dimension based on the Shannon sampling theorem

ABSTRACT

A backward method is proposed to compute the solutions to some class of transport equations at any temporal instant regardless of the dimension. The widely adopted Shannon sampling in information theory and signal processing is employed for the reconstruction of solutions through truncated cardinal series, citing its properties of accuracy in approximation and convenience in construction. With the method of characteristics, approximation coefficients at sampling nodes are obtained via backward tracking along the characteristics. This approach, due to Gobbi et al. [Numerical solution of certain classes of transport equations in any dimension by Shannon sampling, J. Comput. Phys. 229 (2010), pp. 3502–5322], can be considered as either a spectral or a wavelet method. The proposed method is further extended to a backward–forward scheme to solve Cauchy problems by employing a forward evolution along the characteristics. Numerical experiments are presented to verify the effectiveness, efficiency and high accuracy of the proposed method.

KEYWORDS:

• Backward method
• Cauchy problem
• characteristics
• Shannon wavelets
• transport equation

2010 AMS SUBJECT CLASSIFICATIONS:

• 35L04
• 65M25
• 65M70
• 65T60

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments which lead to significant improvement of the paper. The first author would like to thank Prof. T. John Koo in Hong Kong Applied Science and Technology Research Institute (ASTRI) and Prof. Shengzhong Feng in Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences for their constructive advices and kind support.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Qiang Wu  http://orcid.org/0000-0003-0878-6485

Notes

1. Notice that the point $x_p$ is calculated with numerical methods, and has a numerical error at ${10}^{-7}$.

Additional information

Funding

The work was partially supported by National Natural Science Foundation of China [grant numbers 61170307 and 11401563], by China Postdoctoral Science Foundation [grant numbers 2014M560252 and 2015T80333], and by Basic Research Program of Shenzhen [grant number JCYJ20140904154741850].