# American Institute of Mathematical Sciences

September  2021, 17(5): 2519-2526. doi: 10.3934/jimo.2020080

## A novel Chebyshev-collocation spectral method for solving the transport equation

 1 Business School, Shandong Normal University, Jinan, 250014, P.R. China 2 School of Automation and Electrical Engineering, and Key Laboratory of complex Systems and Intellignet Computing, Linyi 276005, Shandong, P.R. China 3 Hubei Key Laboratory of Advanced Control and Intelligent, Automation of Complex Systems, and Engineering Research Center, of Intelligent Geodetection Technology Ministry of Education, China University of Geosciences, Wuhan, 430074, P.R. China

* Corresponding authors: X. Y. Chen (cxy8305@163.com) and T. S. Xia (tsxia@sina.com)

Received  July 2019 Revised  February 2020 Published  September 2021 Early access  April 2020

Fund Project: The authors would like to thank Professor Jianwei Zhou for his works on numerical discretized formulae and tests

In this paper, we employ an efficient numerical method to solve transport equations with given boundary and initial conditions. By the weighted-orthogonal Chebyshev polynomials, we design the corresponding basis functions for spatial variables, which guarantee the stiff matrix is sparse, for the spectral collocation methods. Combining with direct algebraic algorithms for the sparse discretized formula, we solve the equivalent scheme to get the numerical solutions with high accuracy. This collocation methods can be used to solve other kinds of models with limited computational costs, especially for the nonlinear partial differential equations. Some numerical results are listed to illustrate the high accuracy of this numerical method.

Citation: Zhonghui Li, Xiangyong Chen, Jianlong Qiu, Tongshui Xia. A novel Chebyshev-collocation spectral method for solving the transport equation. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2519-2526. doi: 10.3934/jimo.2020080
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##### References:
The maximum errors of $u-u_N$ with log10 at $t = 0.5$
The maximum errors of $u-u_N$ with log10 at $t = 1$
The $L^\infty$-error of numerical solutions at $t = 0.5$
 N CCSM FDM $8$ 2.58952e-4 7.92233e-1 $10$ 3.51652e-6 5.35228e-1 $12$ 2.93379e-7 3.71949e-2 $14$ 4.67534e-9 2.68015e-2 $16$ 2.5433e-2 9.58506e-2
 N CCSM FDM $8$ 2.58952e-4 7.92233e-1 $10$ 3.51652e-6 5.35228e-1 $12$ 2.93379e-7 3.71949e-2 $14$ 4.67534e-9 2.68015e-2 $16$ 2.5433e-2 9.58506e-2
The $L^\infty$-error of numerical solutions at $t = 1$
 N CCSM FDM $8$ 2.99237e-4 8.00453e-1 $10$ 7.33715e-7 5.56804e-1 $12$ 1.66371e-9 4.01949e-2 $14$ 9.97109e-12 3.08050e-2 $16$ 5.74238e-14 1.00513e-2
 N CCSM FDM $8$ 2.99237e-4 8.00453e-1 $10$ 7.33715e-7 5.56804e-1 $12$ 1.66371e-9 4.01949e-2 $14$ 9.97109e-12 3.08050e-2 $16$ 5.74238e-14 1.00513e-2
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