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

Zhonghui Li; Xiangyong Chen; Jianlong Qiu; Tongshui Xia

doi:10.3934/jimo.2020080

Article Contents

2021, Volume 17, Issue 5: 2519-2526. Doi: 10.3934/jimo.2020080

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

Received: July 2019

Revised: February 2020

Early access: April 2020

Published: September 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 65D15, 65L99; Secondary: 34A45.

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Figure 1. The maximum errors of $ u-u_N $ with log10 at $ t = 0.5 $

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Figure 2. The maximum errors of $ u-u_N $ with log10 at $ t = 1 $

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Table 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

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Table 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

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