Spectral collocation solutions to systems of boundary layer type

Abstract

Three spectral collocation methods, namely Laguerre collocation (LC), Laguerre Gauss Radau collocation (LGRC) and mapped Chebyshev collocation (ChC) are used in order to solve some challenging systems of boundary layer problems of third and second orders.

The last two methods enable a Fourier type analysis, mainly (fast) polynomial transformations, which can be used in order to improve the process of optimization of the scaling parameters.

Generally, the second method mentioned above produces the best results. Unfortunately they remain sub geometric with respect to the accuracy.

However, all methods avoid domain truncation and rather arbitrary shooting techniques. Some challenging problems from fluid mechanics, including non-newtonian fluids are accurately solved.

Authors

Călin-Ioan Gheorghiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Keywords

Laguerre collocation; Laguerre Gauss Radau collocation; Fourier-Laguerre analysis; Falkner-Skan; nonlinear heat transfer; rotating disk; Non-Newtonian fluid.

References

See the expanding block below.

Cite this paper as

C.I. Gheorghiu, Spectral collocation solutions to systems of boundary layer type, Numer. Algor., 73 (2016) no. 1, pp 1–14, doi: 10.1007/s11075-015-0083-6

PDF

?

About this paper

Journal

Numerical Algorithms

Publisher Name

Springer

Print ISSN

1017-1398

Online ISSN

1572-9265

Google Scholar Profile

google scholar link

References

References

1. Benacchio, T., Bonaventura, L.: Absorbing boundary conditions: a spectral collocation approach. Int.
J. Numer. Meth. Fluids (2013). doi:10.1002/fld.3768
2. Bernardy, C., Maday, Y. In: Ciarlet, P., Lions, J.L. (eds.): Spectral methods, vol. 5 (Part 2). NorthHolland
(1997)
3. Boyd, J.P.: Chebyshev and fourier spectral methods. Dover Publications, New-York (2000)
4. Boyd, J.P., Rangan, C., Bucksbaum, P.H.: Pseudospectral methods on a semi-infinite interval with
applications to the hydrogen atom: a comparison of the mapped Fourier-sine method with Laguerre
series and rational Chebyshev expansions. J. Comput. Phys. 188, 56–74 (2003)
5. Boyd, J.P.: Chebyshev spectral methods and the lane-emden problem. Numer. Math. Theor. Meth.
Appl. 4, 142–157 (2011)
6. Dragomirescu, I.F., Gheorghiu, C.I.: Analytical and numerical solutions to an electrohydrodynamic
stability problem. Appl. Math. Comput. 216, 3718–3727 (2010). doi:10.1016/j.amc.2010.05.028
7. Fazio, R.: A novel approach to the numerical solution of boundary value problems on infinite intervals.
SIAM J. Numer. Anal. 33, 1473–1483 (1996)
8. Gheorghiu, C.I.: Spectral methods for differential problems. Casa Cartii de Stiinta Publishing House,
Cluj-Napoca, Romania (2007)
9. Gheorghiu, C.I., Dragomirescu, I.F.: Spectral methods in linear stability. Application to thermal
convection with variable gravity field. Appl. Numer. Math 59, 1290–1302 (2009)
10. Gheorghiu, C.I., Hochstenbach, M.E., Plestenjak, B., Rommes, J.: Spectral collocation solutions
to multiparameter Mathieu’s system. Appl. Math. Comput 218, 11990–12000 (2012).
doi:10.1016/j.acm.2012.05.068
11. Gheorghiu, C.I.: Laguerre collocation solutions to boundary layer type problems. Numer. Algor 64,
385–401 (2013). doi:10.1007/s11075-012-9670-y
12. Gheorghiu, C.I.: Pseudospectral solutions to some singular nonlinear BVPs. Applications in nonlinear
mechanics. Numer. Algor. 68, 1–14 (2015). doi:10.1007/s11075-014-9834-z
13. Gheorghiu, C.I.: Spectral Methods for Non-Standard Eigenvalue Problems. Fluid and Structural
Mechanics and Beyond. Springer Cham Heidelberg New York Dordrecht London (2014)
14. Gottlieb, D., Orszag, S.A.: Numerical analysis of spectral methods: theory and applications. SIAM,
Philadelphia (1977)
15. Hoepffner, J.: Implementation of boundary conditions. http://www.lmm.jussieu.fr/∼hoepffner/
research/realizing.pdf, Accessed 2 Feb 2015 (2010)
16. Iacono, R., Boyd, J.P.: The Kidder Equation: uxx + 2xux /
√1 − αu = 0. Stud. Appl. Math. 135, 63–
85 (2014)
17. Liao, S.-J.: A challenging nonlinear problem for numerical techniques. J. Comput. Appl. Math. 181,
467–472 (2005)
18. Liao, S.-J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate.
Int. J. Heat Mass Transfer 48, 2529–2539 (2005)
19. Liao, S.-J.: A new branch of solutions of boundary-layer flows over a permeable stretching plate. Int.
J. Nonlinear Mech. 42, 819–830 (2007)
20. Magyari, E., Keller, B.: Heat transfer characteristics of the separation boundary flow induced by a
continuous stretching surface. J. Phys. D: Appl. Phys. 32, 2876–2881 (1999)
21. Magyari, E., Keller, B.: Exact solutions for self-similar boundary-layer flows induced by permeable
stretching walls. Eur. J. Mech. B – Fluids 19, 109–122 (2000)
22. Mandelzweig, V.B., Tabakin, F.: Quasilinearization approach to nonlinear problems in physics with
applications to nonlinear ODEs. Comput. Phys. Commun. 141, 268–281 (2001)
23. Ockendon, H., Ockendon, J.R.: Viscous flow. Cambridge University Press, Cambridge (1995)
24. O’Neil, M.E., Chorlton, F.: Viscous and compressible fluid dynamics, p. 395. Wiley, New York
(1989)
25. Pantokratoras, A., Fang, T.: Blasius flow with non-linear Rosseland thermal radiation. Meccanica 49,
1539–1545 (2014)
26. Plestenjak, B., Gheorghiu, C.I., Hochstenbach, M.E.: Spectral collocation for multiparameter eigenvalue
problems arising from separable boundary value problems. J. Comput. Phys. 298, 585–601
(2015). doi:10.1016/j.jcp.2015.06.015
27. Rosales-Vera, M., Valencia, A.: Solutions of Falkner-Skan equation with heat transfer by Fourier
series. Int. Commun. Heat Mass 37, 761–765 (2010)
28. Shen, J., Tang, T., Wang, L.-L.: Spectral methods. Algorithms, analysis and applications. Springer,
Berlin (2011)
29. Wang, L.-L.: Discrete transform of Laguerre function approach. http://www.ntu.edu.sg/home/lilian/
book.htm, Accessed 5 May 2014 (2011)
30. Weideman, J.A.C., Reddy, S.C.: A MATLAB differentiation matrix suite. ACM Trans. Math.
Software 26, 465–519 (2000)
31. von Winckel, G.: Fast Chebyshev Transform. http://www.mathworks.com/matlabcentral/
fileexchange/4591-fast-chebyshev-transform–1d-, Accessed 15 May 2015 (2004)

Related Posts

Menu