Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Asymptotic approximation of singularly perturbed convection-diffusion problems with discontinuous derivatives of the Dirichlet data

Authors: José L. López and Ester Pérez Sinusía
Journal: Quart. Appl. Math. 63 (2005), 527-543
MSC (2000): Primary 35C20, 41A60
DOI: https://doi.org/10.1090/S0033-569X-05-00962-0
Published electronically: August 17, 2005
MathSciNet review: 2169032
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a singularly perturbed convection-diffusion equation, $-\epsilon\bigtriangleup u +\overrightarrow v\cdot\overrightarrow\nabla u=0$, defined on two domains: a quarter plane, $(x,y)\in(0,\infty)\times(0,\infty)$, and a half plane, $(x,y)\in(-\infty,\infty)\times(0,\infty)$. We consider for these problems Dirichlet boundary conditions with discontinuous derivatives at some points of the boundary. We obtain for each problem an exact representation of the solution in the form of an integral. From this integral we derive an asymptotic expansion of the solution when the singular parameter $\epsilon\to 0^+$ (with fixed distance $r$ to the points of discontinuity of the boundary condition). It is shown that, in both problems, the first term of the expansion contains the primitive of an error function. This term characterizes the effect of the discontinuities on the $\epsilon-$behaviour of the solution and its derivatives in the boundary or internal layers.

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  • 1. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970.
  • 2. C. Clavero, J. C. Jorge and F. Lisbona, Uniformly convergent schemes for singular perturbation problems combining alternating directions and exponential fitting techniques, Boole Press, Ireland, 1993, pp. 33-52. MR 1245731 (94j:65106)
  • 3. L. P. Cook and G. S. S. Ludford, The behavior as $\epsilon \to 0^+$ of solutions to $\epsilon \nabla ^2w=\partial w/\partial y$ on the rectangle $0\le x\le l$, $\vert y\vert \le 1$, SIAM J. Math. Anal., 4, no. 1 (1973) 161-184. MR 0364828 (51:1082)
  • 4. W. Eckhaus, Matched Asymptotic Expansions and Singular Perturbations, North-Holland, Amsterdam, 1973. MR 0670800 (58:32369)
  • 5. W. Eckhaus and E. M. de Jager, Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type, Arch. Rat. Mech. Anal., 23, (1966) 26-86. MR 0206464 (34:6283)
  • 6. J. Grasman, On singular perturbations and parabolic boundary layers, J. Eng. Math., 2, no. 2 (1968) 163-172. MR 0233059 (38:1382)
  • 7. J. Grasman, An elliptic singular perturbation problem with almost characteristic boundaries, J. Math. Anal. Appl., 46, (1974) 438-446. MR 0352672 (50:5159)
  • 8. G. W. Hedstrom and A. Osterheld, The effect of cell Reynolds number on the computation of a boundary layer, J. Comput. Phys., 37 (1980) 399-421. MR 0588260 (82a:76023)
  • 9. A.M. Il'in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, AMS, Providence, 1992. MR 1182791 (93g:35016)
  • 10. J. Kevorkian and J.D. Cole, Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York, 1996. MR 1392475 (97k:34001)
  • 11. J.L. López and E. Pérez Sinusía, Asymptotic expansions for two singularly perturbed convection-diffusion problems with discontinuous data: the quarter plane and the infinite strip. Stud. Appl. Math., 113, (2004) 57-89. MR 2061650
  • 12. J.L. López and E. Pérez Sinusía, Analytic approximations for a singularly perturbed convection-diffusion problems with discontinuous data in a half-infinite strip. Acta Appl. Math., 82, (2004) 101-117. MR 2061479 (2005e:35048)
  • 13. J. H. Miller, E. O'Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, 1996. MR 1439750 (98c:65002)
  • 14. R. E. O'Malley, Introduction to Singular Perturbations, Academic Press, New York, 1974.MR 0402217 (53:6038)
  • 15. S.-D. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18, no. 5 (1987) 1467-1511. MR 0902346 (88j:35020)
  • 16. S.-D. Shih, A Novel uniform expansion for a singularly perturbed parabolic problem with corner singularity, Meth. Appl. Anal., 3, no. 2 (1996) 203-227. MR 1420366 (98c:35011)
  • 17. N.M. Temme, Analytical methods for a singular perturbation problem. The quarter plane, C.W.I. Report, 125, (1971).
  • 18. N.M. Temme, Analytical methods for a singular perturbation problem in a sector, SIAM J. Math. Anal., 5, no. 6 (1974) 876-887. MR 0369876 (51:6105)
  • 19. N.M. Temme, Analytical methods for a selection of elliptic singular perturbation problems, Recent advances in differential equations (Kunming, 1997), 131-148, Pitman Res. Notes Math. Ser., 386, Longman, Harlow, 1998. MR 1628077 (99c:35018)
  • 20. A. B. Vasil'eva, V. F. Butuzov and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM, Philadelphia, 1995. MR 1316892 (96a:34119)
  • 21. C. H. Ou and R. Wong, On a two-point boundary value problem with spurious solution. Stud. Appl. Math. 111 no. 4 (2003) 377-408. MR 2004239 (2004f:34081)

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Additional Information

José L. López
Affiliation: Departamento de Matemática e Informática, Universidad Pública de Navarra, 31006-Pamplona, Spain
Email: jl.lopez@unavarra.es

Ester Pérez Sinusía
Affiliation: Departamento de Matemática e Informática, Universidad Pública de Navarra, 31006-Pamplona, Spain
Email: ester.perez@unavarra.es

DOI: https://doi.org/10.1090/S0033-569X-05-00962-0
Keywords: Singular perturbation problem, discontinuous boundary data, asymptotic expansion, error function
Received by editor(s): October 28, 2004
Published electronically: August 17, 2005
Article copyright: © Copyright 2005 Brown University

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