A backward approach to certain class of transport equations in any dimension based on the Shannon sampling theorem. Issue 10 (3rd October 2017)

• Record Type: Journal Article
• Authors: Wu, Qiang; Shi, Feng
• Is Part Of: International journal of computer mathematics. Volume 94:Issue 10(2017)
• Page Start: 1943
• Page End: 1967
• Publication Date: 2017-10-03
• Subjects: Backward method -- Cauchy problem -- characteristics -- Shannon wavelets -- transport equation; 35L04 -- 65M25 -- 65M70 -- 65T60; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2016.1274739
• Languages: English
• ISSNs: 0020-7160

Abstract

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.