Shifted-Legendre orthonormal method for high-dimensional heat conduction equations

Liangcai Mei; Boying Wu; Yingzhen Lin; Liangcai Mei; Boying Wu; Yingzhen Lin

doi:10.3934/math.2022525

AIMS Mathematics

2022, Volume 7, Issue 5: 9463-9478. doi: 10.3934/math.2022525

Previous Article Next Article

Research article Special Issues

Shifted-Legendre orthonormal method for high-dimensional heat conduction equations

1.
Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China
2.
Zhuhai Campus, Beijing Institute of Technology, Zhuhai, Guangdong, 519088, China

Received: 01 October 2021 Revised: 02 March 2022 Accepted: 07 March 2022 Published: 14 March 2022
MSC : 65M12, 65N12

In this paper, a numerical alogorthm for solving high-dimensional heat conduction equations is proposed. Based on Shifted-Legendre orthonormal polynomial and $ \varepsilon- $best approximate solution, we extend the algorithm from low-dimensional space to high-dimensional space, and prove the convergence of the algorithm. Compared with other numerical methods, the proposed algorithm has the advantages of easy expansion and high convergence order, and we prove that the algorithm has $ \alpha $-Order convergence. The validity and accuracy of this method are verified by some numerical experiments.
- shifted-Legendre polynomials,
- heat conduction equation,
- $ \varepsilon- $best approximate solution,
- convergence order
Citation: Liangcai Mei, Boying Wu, Yingzhen Lin. Shifted-Legendre orthonormal method for high-dimensional heat conduction equations[J]. AIMS Mathematics, 2022, 7(5): 9463-9478. doi: 10.3934/math.2022525

Related Papers:

Abstract

In this paper, a numerical alogorthm for solving high-dimensional heat conduction equations is proposed. Based on Shifted-Legendre orthonormal polynomial and $ \varepsilon- $best approximate solution, we extend the algorithm from low-dimensional space to high-dimensional space, and prove the convergence of the algorithm. Compared with other numerical methods, the proposed algorithm has the advantages of easy expansion and high convergence order, and we prove that the algorithm has $ \alpha $-Order convergence. The validity and accuracy of this method are verified by some numerical experiments.

References

[1]	J. Lei, Q. Wang, X. Liu, Y. Gu, C. Fan, A novel space-time generalized FDM for transient heat conduction problems, Eng. Anal. Bound. Elem., 119 (2020), 1–12. https://doi.org/10.1016/j.enganabound.2020.07.003 doi: 10.1016/j.enganabound.2020.07.003
[2]	C. Ku, C. Liu, W. Yeih, C. Liu, C. Fan, A novel space-time meshless method for solving the backward heat conduction problem, Int. J. Heat Mass Tran., 130 (2019), 109–122. https://doi.org/10.1016/j.ijheatmasstransfer.2018.10.083 doi: 10.1016/j.ijheatmasstransfer.2018.10.083
[3]	L. Qiu, W. Chen, F. Wang, C. Liu, Q. Hu, Boundary function method for boundary identification in two-dimensional steady-state nonlinear heat conduction problems, Eng. Anal. Bound. Elem., 103 (2019), 101–108. https://doi.org/10.1016/j.enganabound.2019.03.004 doi: 10.1016/j.enganabound.2019.03.004
[4]	Z. She, K. Wang, P. Li, Hybrid Trefftz polygonal elements for heat conduction problems with inclusions/voids, Comput. Math. Appl., 78 (2019), 1978–1992. https://doi.org/10.1016/j.camwa.2019.03.032 doi: 10.1016/j.camwa.2019.03.032
[5]	J. T. Oden, I. Babuska, C. E. Baumann, A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., 146 (1998), 491–519. https://doi.org/10.1006/jcph.1998.6032 doi: 10.1006/jcph.1998.6032
[6]	W. Kanjanakijkasem, A finite element method for prediction of unknown boundary conditions in two-dimensional steady-state heat conduction problems, Int. J. Heat Mass Tran., 88 (2015), 891–901. https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.019 doi: 10.1016/j.ijheatmasstransfer.2015.05.019
[7]	S. Clain, G. J. Machado, J. M. Nóbrega, R. M. S. Pereira, A sixth-order finite volume method for multidomain convection-diffusion problems with discontinuous coefficients, Comput. Method. Appl. Mech. Eng., 267 (2013), 43–64. https://doi.org/10.1016/j.cma.2013.08.003 doi: 10.1016/j.cma.2013.08.003
[8]	G. Manzini, A. Russo, A finite volume method for advection-diffusion problems in convection-dominated regimes, Comput. Method. Appl. Mech. Eng., 197 (2008), 1242–1261. https://doi.org/10.1016/j.cma.2007.11.014 doi: 10.1016/j.cma.2007.11.014
[9]	L. Mei, H. Sun, Y. Lin, Numerical method and convergence order for second-order impulsive differential equations, Adv. Differ Equ., 2019 (2019), 260. https://doi.org/10.1186/s13662-019-2177-2 doi: 10.1186/s13662-019-2177-2
[10]	L. Mei, A novel method for nonlinear impulsive differential equations in broken reproducing Kernel space, Acta Math. Sci., 40 (2020), 723–733. https://doi.org/10.1007/s10473-020-0310-7 doi: 10.1007/s10473-020-0310-7
[11]	L. Mei, Y. Jia, Y. Lin, Simplified reproducing kernel method for impulsive delay differential equations, Appl. Math. Lett., 83 (2018), 123–129. https://doi.org/10.1016/j.aml.2018.03.024 doi: 10.1016/j.aml.2018.03.024
[12]	M. Xu, E. Tohidi, A Legendre reproducing kernel method with higher convergence order for a class of singular two-point boundary value problems, J. Appl. Math. Comput., 67 (2021), 405–421. https://doi.org/10.1007/s12190-020-01494-6 doi: 10.1007/s12190-020-01494-6
[13]	M. Xu, L. Zhang, E. Tohidi, A fourth-order least-squares based reproducing kernel method for one-dimensional elliptic interface problems, Appl. Numer. Math., 162 (2021), 124–136. https://doi.org/10.1016/j.apnum.2020.12.015 doi: 10.1016/j.apnum.2020.12.015
[14]	M. Xu, J. Niu, E. Tohidi, J. Hou, D. Jiang, A new least-squares-based reproducing kernel method for solving regular and weakly singular Volterra-Fredholm integral equations with smooth and nonsmooth solutions, Math. Method. Appl. Sci., 44 (2021), 10772–10784. https://doi.org/10.1002/mma.7444 doi: 10.1002/mma.7444
[15]	X. H. Wu, S. P. Shen, W. Q. Tao, Meshless local Petrov-Galerkin collocation method for two-dimensional heat conduction problems, CMES, 22 (2007), 65–76.
[16]	Y. Zhang, X. Zhang, C. W. Shu, Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-diffusion equations on triangular meshes, J. Comput. Phys., 234 (2013), 295–316. https://doi.org/10.1016/j.jcp.2012.09.032 doi: 10.1016/j.jcp.2012.09.032
[17]	Y. Cheng, C. W. Shu, Superconvergence of local discontinuous Galerkin methods for one-dimensional convection-diffusion equations, Comput. Struct., 87 (2009), 630–641. https://doi.org/10.1016/j.compstruc.2008.11.012 doi: 10.1016/j.compstruc.2008.11.012
[18]	S. Jun, D. W. Kim, Axial Green's function method for steady Stokes flow in geometrically complex domains, J. Comput. Phys., 230 (2011), 2095–2124. https://doi.org/10.1016/j.jcp.2010.12.007 doi: 10.1016/j.jcp.2010.12.007
[19]	W. Lee, D. W. Kim, Localized axial Green's function method for the convection-diffusion equations in arbitrary domains, J. Comput. Phys., 275 (2014), 390–414. https://doi.org/10.1016/j.jcp.2014.06.050 doi: 10.1016/j.jcp.2014.06.050
[20]	M. Xu, A high order scheme for unsteady heat conduction equations, Appl. Math. Comput., 384 (2019), 565–574. https://doi.org/10.1016/j.amc.2018.12.024 doi: 10.1016/j.amc.2018.12.024
[21]	H. W. Sun, L. Z. Li, A CCD-ADI method for unsteady convection-diffusion equations, Comput. Phys. Commun., 185 (2014), 790–797. https://doi.org/10.1016/j.cpc.2013.11.009 doi: 10.1016/j.cpc.2013.11.009
[22]	C. Wang, F. Wang, Y. Gong, Analysis of 2D heat conduction in nonlinear functionally graded materials using a local semi-analytical meshless method, AIMS Mathematics, 6 (2021), 12599–12618. https://doi.org/10.3934/math.2021726 doi: 10.3934/math.2021726
[23]	F. Wang, C. Wang, Z. Chen, Local knot method for 2D and 3D convection-diffusion-reaction equations in arbitrary domains, Appl. Math. Lett., 105 (2020), 106308. https://doi.org/10.1016/j.aml.2020.106308 doi: 10.1016/j.aml.2020.106308
[24]	X. Yue, F. Wang, Q. Hua, X. Qiu, A novel space-time meshless method for nonhomogeneous convection-diffusion equations with variable coefficients, Appl. Math. Lett., 92 (2019), 144–150. https://doi.org/10.1016/j.aml.2019.01.018 doi: 10.1016/j.aml.2019.01.018
[25]	F. Wang, C. Fan, C. Zhang, J. Lin, A localized space-time method of fundamental solutions for diffusion and convection-diffusion problems, Adv. Appl. Math. Mech., 12 (2020), 940–958. https://doi.org/10.4208/aamm.OA-2019-0269 doi: 10.4208/aamm.OA-2019-0269
[26]	H. Sun, L. Mei, Y. Lin, A new algorithm based on improved Legendre orthonormal basis for solving second-order BVPs, Appl. Math. Lett., 112 (2021), 106732. https://doi.org/10.1016/j.aml.2020.106732 doi: 10.1016/j.aml.2020.106732
[27]	M. U. Rehman, R. A. Khan, The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci., 16 (2011), 4163–4173. https://doi.org/10.1016/j.cnsns.2011.01.014 doi: 10.1016/j.cnsns.2011.01.014
[28]	S. Sheikhi, M. Matinfar, M. A. Firoozjaee, Numerical solution of variable-order differential equations via the Ritz-approximation Method by shifted Legendre polynomials, Int. J. Appl. Comput. Math., 7 (2021), 22. https://doi.org/10.1007/s40819-021-00962-2 doi: 10.1007/s40819-021-00962-2
[29]	C. G. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods: fundamentals in single domains, Berlin: Springer, 2006. https://doi.org/10.1007/978-3-540-30726-6
[30]	B. Wu, Y. Lin, Application oriented the reproducing Kernel space, Beijing: Beijing Science Press, 2012.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)