Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions

Öz

This paper aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter $\varepsilon$. The approximation is based on a couple of methods: A spectral method of Galerkin type with a basis composed from Legendre-polynomials and a Gauss quadrature of type Gauss-Lobatto applied for integral calculations with a stability and convergence analysis. In addition, a Crank-Nicolson scheme is used for temporal solution as a finite difference method. Several numerical examples are discussed to show the efficiency of the proposed numerical method, specially when $\varepsilon$ tends to zero so that we obtain the exact solution of the classic problem with homogeneous Dirichlet boundary conditions. The numerical convergence is well presented in different examples. Therefore, we build an efficient numerical method for different types of partial differential equations with different boundary conditions.

Anahtar Kelimeler

• Spectral method
• Galerkin
• Robin's conditions
• Advection-Diffusion equation
• Gauss-Quadrature
• Difference scheme
• Crank Nicolson scheme.

Kaynakça

1. Isenberg, J and Gutfinger, C, Heat transfer to a draining film, 1nt. J. Heat Trans 16, 505--512, 1973.
2. Parlarge, JY, Water transport in soils, Ann. Rev. Fluids Mech 2, 77--102, 1980.
3. Chatwin, PC and Allen, CM, Mathematical models of dispersion in rivers and estuaries, Ann. Rev. Fluids Mech 17, 119--149, 1985.
4. Kumar, N, Unsteady flow against dispersion in finite porous media, J Hydrol 63, 345--358, 1998.
5. Mojtabi, A and Deville, MO, One-dimensional linear advection–diffusion equation: Analytical and finite element solutions, Computers \& Fluids 107, 189--195, 2015.
6. Hutomo, GD and Kusuma, J and Ribal, A and Mahie, AG and Aris, N, Numerical solution of 2-d advection-diffusion equation with variable coefficient using du-fort frankel method, J Physics: Conf. Series 1180, 012009, 2019.
7. Abdelhamid, T and Elsheikh, AH and Elazab, A and Sharshir, SW and Selima, ES and Jiang, D, Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems, Inverse Problems in Science and Engineering 26(9), 1213--1248, 2018.
8. Abdelhamid, T and Elsheikh, AH and Omisore, OM and Saeed, NA and Muthuramalingam, T and Chen, R and Alam, MM, Reconstruction of the heat transfer coefficients and heat fluxes in heat conduction problems, Mathematics and Computers in Simulation 187, 134--154, 2021.

9. Abdelhamid, T and Chen, R and Alam, MM, Nonlinear conjugate gradient method for identifying Young's modulus of the elasticity imaging inverse problem, Inverse Problems in Science and Engineering 29(12), 2165--2185,2021.
10. Ma, X and Wang, Y and Zhu, X and Liu, W and Xiao, W and Lan, Q, A High-Efficiency Spectral Method for Two-Dimensional Ocean Acoustic Propagation Calculations, Entropy 23(9), 1227, 2021.
11. Balyan, LK and Mittal, AK and Kumar, M and Choube, M, Stability analysis and highly accurate numerical approximation of fisher’s equations using pseudospectral method, Mathematics and Computers in Simulation 117, 86--104, 2020.
12. Jedrzejewski, F, Introduction aux m\'ethodes num\'eriques. Deuxi\`eme \'edition, Springer-Verlag, Paris, France, 2005.
13. Quarteroni, A and Canuto, C and Hussaini, MY and Zang, TA, Spectral methods in fluid dynamics, Springer-Verlag, Berlin, Heidelberg, 1988.
14. Shen, J and Tang, T and Wang, L, Spectral methods, algorithms, analysis and applications, Springer-Verlag, Berlin, Heidelberg, 2011.
15. Gottlieb, D and Orszag, S, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, 1977.
16. Carpenter, MH and Gottlieb, D, Spectral methods on arbitrary grids, Journal of Computational Physics 129(1), 74--86, 1996.
17. Funaro, D, Polynomial approximation of differential equations, Springer-Verlag, Berlin, Heidelberg,1992.
18. Carlson, BC, Special functions of applied mathematics, Academic Press, New York, 1978
19. Quarteroni, A and Sacco, R and Saleri, F, M\'ethodes num\'eriques. Algorithmes, analyse et application, Springer-Verlag, Milano, Italia, 2004.
20. Chattouh, A and Saoudi, K, Error analysis of Legendre-Galerkin spectral method for a parabolic equation with Dirichlet-Type non-local boundary conditions, Mathematical Modelling and Analysis 26(2), 287--303, 2021.
21. Allaire, G, Analyse num\'erique et optimisation, Edition de l'\'ecole polytechnique, 2007.
22. Shen, J, Efficient spectral-Galerkin method I. Direct solvers for the second and fourth order equations using Legendre polynomials, SIAM Journal on Scientific Computing 15, 1489--1505, 1994.
23. Shen, J, Efficient Chebyshev-Legendre Galerkin methods for elliptic problems, Proceedings of the third internatioanl conference on spectral and hight order methods, Houston Journal of Mathematics 70(34), 233--239, 1998.
24. Quarteroni, A and Canuto, C and Hussaini, MY and Zang, TA, Spectral methods, fundamentals in single domains, Springer-Verlag, Berlin Heidelberg, 2006.