# 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 and Management Optimization, 2021, 17 (5) : 2519-2526. doi: 10.3934/jimo.2020080
##### References:
 [1] B. Bialecki, Sinc-collection methods for two-point boundary value problems, Ima Journal of Numerical Analysis, 11 (1991), 357-375.  doi: 10.1093/imanum/11.3.357. [2] A. G. Buchan, C. C. Pain, M. D. Eaton, R. P. Smedley-Stevenson and A. J. H. Goddard, Chebyshev spectral hexahedral wavelets on the sphere for angular discretisations of the boltzmann transport equation, Annals of Nuclear Energy, 35 (2008), 1098-1108.  doi: 10.1016/j.anucene.2007.08.021. [3] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-84108-8. [4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898719208. [5] J. D. Dockery, Numerical solution of travelling waves for reaction-diffusion equations via the sinc-galerkin method, In Bowers K., Lund J. (eds) Computation and Control II. Progress in Systems and Control Theory, 11 (1991), 95-113. [6] M. El-Gamel, A comparison between the Sinc-Galerkin and the modified decomposition methods for solving two-point boundary-value problems, Journal of Computational Physics, 223 (2007), 369-383.  doi: 10.1016/j.jcp.2006.09.025. [7] P. Heidelberger and P. D. Welch, A spectral method for confidence interval generation and run length control in simulations, Communications of the ACM, 24 (1981), 233-245.  doi: 10.1145/358598.358630. [8] A. Ishimaru, Wave propagation and scattering in random media and rough surfaces, Proceedings of the IEEE, 79 (1991), 1359-1366. [9] A. D. Kim and A. Ishimaru, A chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in a discrete random medium, J. Comput. Phys, 152 (1999), 264-280.  doi: 10.1006/jcph.1999.6247. [10] V. B. Kisselev, L. Roberti and G. Perona, An application of the finite element method to the solution of the radiative transfer equation, Journal of Quantitative Spectroscopy and Radiative Transfer, 51 (1994), 603-614.  doi: 10.1016/0022-4073(94)90114-7. [11] A. Lundbladh, D. S. Henningson and A. V. Johansson, An Efficient Spectral Integration Method for the Solution of the Navier-Stokes Equations, Aeronautical Research Institute of Sweden Bromma, 1992. [12] X. J. Li and C. J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM Journal on Numerical Analysis, 47 (2009), 2108-2131.  doi: 10.1137/080718942. [13] A. M. Mao, L. J. Yang, A. X. Qian and S. X. Luan, Existence and concentration of solutions of schrödinger-poisson system, Applied Mathematics Letters, 68 (2017), 8-12.  doi: 10.1016/j.aml.2016.12.014. [14] S. R. Merton, C. C. Pain, R. P. Smedley-Stevenson, A. G. Buchan and M. D. Eaton, Optimal discontinuous finite element methods for the boltzmann transport equation with arbitrary discretisation in angle, Annals of Nuclear Energy, 35 (2008), 1741-1759.  doi: 10.1016/j.anucene.2008.01.023. [15] H. F. Niu, D. P. Yang and J. W. Zhou, Numerical analysis of an optimal control problem governed by the stationary navier-stokes equations with global velocity-constrained, Communications in Computational Physics, 24 (2018), 1477-1502.  doi: 10.4208/cicp.oa-2017-0045. [16] B. Wang, A. Iserles and X. Y. Wu, Arbitrary-order trigonometric fourier collocation methods for multi-frequency oscillatory systems, Foundations of Computational Mathematics, 16 (2016), 151-181.  doi: 10.1007/s10208-014-9241-9. [17] B. Wang, F. W. Meng and Y. L. Fang, Efficient implementation of rkn-type fourier collocation methods for second-order differential equations, Applied Numerical Mathematics, 119 (2017), 164-178.  doi: 10.1016/j.apnum.2017.04.008. [18] B. Wang, X. Y. Wu and F. W. Meng, Trigonometric collocation methods based on lagrange basis polynomials for multi-frequency oscillatory second-order differential equations, Journal of Computational and Applied Mathematics, 313 (2017), 185-201.  doi: 10.1016/j.cam.2016.09.017. [19] B. Wang, H. L. Yang and F. W. Meng, Sixth-order symplectic and symmetric explicit erkn schemes for solving multi-frequency oscillatory nonlinear hamiltonian equations, Calcolo, 54 (2017), 117-140.  doi: 10.1007/s10092-016-0179-y. [20] B. Wang, Triangular splitting implementation of rkn-type fourier collocation methods for second-order differential equations, Mathematical Methods in the Applied Sciences, 41 (2018), 1998-2011.  doi: 10.1002/mma.4727. [21] X. Y. Wu and B. Wang, Exponential fourier collocation methods for solving first-order differential equations, In Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations, Springer, Singapore, (2018), 55–84. [22] J. W. Zhou and D. P. Yang, An improved a posteriori error estimate for the galerkin spectral method in one dimension, Computers & Mathematics with Applications, 61 (2011), 334-340.  doi: 10.1016/j.camwa.2010.11.008. [23] J. W. Zhou, J. Zhang and X. Q. Xing, Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state, Computers & Mathematics with Applications, 72 (2016), 2549-2561.  doi: 10.1016/j.camwa.2016.08.009. [24] J. W. Zhou, J. Zhang, H. T. Xie and Y. Yang, Error estimates of spectral element methods with generalized jacobi polynomials on an interval, Applied Mathematics Letters, 74 (2017), 199-206.  doi: 10.1016/j.aml.2017.03.010. [25] J. W. Zhou, Z. W. Jiang, H. T. Xie and H. F. Niu, The error estimates of spectral methods for 1-dimension singularly perturbed problem, Applied Mathematics Letters, 100 (2020), 106001, 8 pp. doi: 10.1016/j.aml.2019.106001.

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##### References:
 [1] B. Bialecki, Sinc-collection methods for two-point boundary value problems, Ima Journal of Numerical Analysis, 11 (1991), 357-375.  doi: 10.1093/imanum/11.3.357. [2] A. G. Buchan, C. C. Pain, M. D. Eaton, R. P. Smedley-Stevenson and A. J. H. Goddard, Chebyshev spectral hexahedral wavelets on the sphere for angular discretisations of the boltzmann transport equation, Annals of Nuclear Energy, 35 (2008), 1098-1108.  doi: 10.1016/j.anucene.2007.08.021. [3] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-84108-8. [4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898719208. [5] J. D. Dockery, Numerical solution of travelling waves for reaction-diffusion equations via the sinc-galerkin method, In Bowers K., Lund J. (eds) Computation and Control II. Progress in Systems and Control Theory, 11 (1991), 95-113. [6] M. El-Gamel, A comparison between the Sinc-Galerkin and the modified decomposition methods for solving two-point boundary-value problems, Journal of Computational Physics, 223 (2007), 369-383.  doi: 10.1016/j.jcp.2006.09.025. [7] P. Heidelberger and P. D. Welch, A spectral method for confidence interval generation and run length control in simulations, Communications of the ACM, 24 (1981), 233-245.  doi: 10.1145/358598.358630. [8] A. Ishimaru, Wave propagation and scattering in random media and rough surfaces, Proceedings of the IEEE, 79 (1991), 1359-1366. [9] A. D. Kim and A. Ishimaru, A chebyshev spectral method for radiative transfer equations applied to electromagnetic wave propagation and scattering in a discrete random medium, J. Comput. Phys, 152 (1999), 264-280.  doi: 10.1006/jcph.1999.6247. [10] V. B. Kisselev, L. Roberti and G. Perona, An application of the finite element method to the solution of the radiative transfer equation, Journal of Quantitative Spectroscopy and Radiative Transfer, 51 (1994), 603-614.  doi: 10.1016/0022-4073(94)90114-7. [11] A. Lundbladh, D. S. Henningson and A. V. Johansson, An Efficient Spectral Integration Method for the Solution of the Navier-Stokes Equations, Aeronautical Research Institute of Sweden Bromma, 1992. [12] X. J. Li and C. J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM Journal on Numerical Analysis, 47 (2009), 2108-2131.  doi: 10.1137/080718942. [13] A. M. Mao, L. J. Yang, A. X. Qian and S. X. Luan, Existence and concentration of solutions of schrödinger-poisson system, Applied Mathematics Letters, 68 (2017), 8-12.  doi: 10.1016/j.aml.2016.12.014. [14] S. R. Merton, C. C. Pain, R. P. Smedley-Stevenson, A. G. Buchan and M. D. Eaton, Optimal discontinuous finite element methods for the boltzmann transport equation with arbitrary discretisation in angle, Annals of Nuclear Energy, 35 (2008), 1741-1759.  doi: 10.1016/j.anucene.2008.01.023. [15] H. F. Niu, D. P. Yang and J. W. Zhou, Numerical analysis of an optimal control problem governed by the stationary navier-stokes equations with global velocity-constrained, Communications in Computational Physics, 24 (2018), 1477-1502.  doi: 10.4208/cicp.oa-2017-0045. [16] B. Wang, A. Iserles and X. Y. Wu, Arbitrary-order trigonometric fourier collocation methods for multi-frequency oscillatory systems, Foundations of Computational Mathematics, 16 (2016), 151-181.  doi: 10.1007/s10208-014-9241-9. [17] B. Wang, F. W. Meng and Y. L. Fang, Efficient implementation of rkn-type fourier collocation methods for second-order differential equations, Applied Numerical Mathematics, 119 (2017), 164-178.  doi: 10.1016/j.apnum.2017.04.008. [18] B. Wang, X. Y. Wu and F. W. Meng, Trigonometric collocation methods based on lagrange basis polynomials for multi-frequency oscillatory second-order differential equations, Journal of Computational and Applied Mathematics, 313 (2017), 185-201.  doi: 10.1016/j.cam.2016.09.017. [19] B. Wang, H. L. Yang and F. W. Meng, Sixth-order symplectic and symmetric explicit erkn schemes for solving multi-frequency oscillatory nonlinear hamiltonian equations, Calcolo, 54 (2017), 117-140.  doi: 10.1007/s10092-016-0179-y. [20] B. Wang, Triangular splitting implementation of rkn-type fourier collocation methods for second-order differential equations, Mathematical Methods in the Applied Sciences, 41 (2018), 1998-2011.  doi: 10.1002/mma.4727. [21] X. Y. Wu and B. Wang, Exponential fourier collocation methods for solving first-order differential equations, In Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations, Springer, Singapore, (2018), 55–84. [22] J. W. Zhou and D. P. Yang, An improved a posteriori error estimate for the galerkin spectral method in one dimension, Computers & Mathematics with Applications, 61 (2011), 334-340.  doi: 10.1016/j.camwa.2010.11.008. [23] J. W. Zhou, J. Zhang and X. Q. Xing, Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state, Computers & Mathematics with Applications, 72 (2016), 2549-2561.  doi: 10.1016/j.camwa.2016.08.009. [24] J. W. Zhou, J. Zhang, H. T. Xie and Y. Yang, Error estimates of spectral element methods with generalized jacobi polynomials on an interval, Applied Mathematics Letters, 74 (2017), 199-206.  doi: 10.1016/j.aml.2017.03.010. [25] J. W. Zhou, Z. W. Jiang, H. T. Xie and H. F. Niu, The error estimates of spectral methods for 1-dimension singularly perturbed problem, Applied Mathematics Letters, 100 (2020), 106001, 8 pp. doi: 10.1016/j.aml.2019.106001.
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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