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B-spline collocation method for the singular-perturbation problem using artificial viscosity. (English) Zbl 1165.65371
Summary: We develop a B-spline collocation method using artificial viscosity for solving singularly-perturbed equations given by $\epsilon u''(x)+a(x)u'(x)+b(x)u(x)=f(x)$, $a(x)\ge a^*>0$, $b(x)\ge b^*>0$, $u(0)=\alpha$, $u(1)=\beta$. We use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh and use B-spline collocation method which leads to a tridiagonal linear system. The convergence analysis is given and the method is shown to have uniform convergence of second order. The design of artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behaviour of the method with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution.

65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
Full Text: DOI
[1] Cole, J.: Perturbation methods in applied mathematics, (1968) · Zbl 0162.12602
[2] Eckhaus, W.: Matched asymptotic expansions and singular perturbations, North-holland mathematics studies 6 (1973) · Zbl 0255.34002
[3] Jr., R. E. O’malley: Introduction to singular perturbation, (1979)
[4] Flaherty, J. E.; Mathon, W.: Collocation with polynomial and tension splines for singularly-perturbed boundary value problems, SIAM J. Sci. stat. Comput. 1, No. 2, 260-289 (1980) · Zbl 0465.65045 · doi:10.1137/0901018
[5] Jain, M. K.; Aziz, T.: Numerical solution of stiff and convection diffusion equations using adaptive spline function approximation, Appl. math. Modelling 7, 57-62 (1983) · Zbl 0512.65083 · doi:10.1016/0307-904X(83)90163-4
[6] Surla, K.; Stojanovic, M.: Solving singularly perturbed boundary-value problems by spline in tension, J. comput. Appl. math. 24, 355-363 (1988) · Zbl 0664.65081 · doi:10.1016/0377-0427(88)90297-X
[7] Abrahamson, L. R.; Keller, H. B.; Kreiss, H. O.: Difference approximations for singular perturbations of systems of ordinary differential equations, Numer. math. 22, 367-391 (1974) · Zbl 0314.65042 · doi:10.1007/BF01436920
[8] A.E. Berger, J.M. Solomon, M. Ciment, Higher order accurate tridiagonal difference methods for diffusion convection equations, in: Proceedings of the Third IMACS Conference on Computer Methods for Partial Differential Equations, Lehigh University, 1979
[9] Flaherty, J. E.; Jr., R. E. O’malley: The numerical solution of boundary value problems for stiff differential equations, Math. comp. 31, 66-93 (1977) · Zbl 0402.65049 · doi:10.2307/2005781
[10] O’riordan, E.; Stynes, M.: A uniformly accurate finite element method for a singularly perturbed one-dimensional reaction-diffusion problem, Math. comp. 47, 555-570 (1986) · Zbl 0625.65073 · doi:10.2307/2008172
[11] Schatz, A. H.; Wahlbin, L. B.: On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. comp. 40, No. 161, 47-89 (1983) · Zbl 0518.65080 · doi:10.2307/2007363
[12] Stynes, M.; O’riordan, E.: A finite element method for a singularly perturbed boundary value problem, Numer. math. 50, 1-15 (1986) · Zbl 0583.65054 · doi:10.1007/BF01389664
[13] Sakai, M.; Usmani, R. A.: A class of simple exponential B-splines and their applications to numerical solution to singular perturbation problems, Numer. math. 65, 493-500 (1989) · Zbl 0676.65088 · doi:10.1007/BF01398912
[14] Prenter, P. M.: Splines and variational methods, (1989) · Zbl 0718.65053
[15] Ahlberg, J. H.; Nilson, E. N.; Walsh, J. L.: The theory of splines and their applications, (1967) · Zbl 0158.15901
[16] Doolan, E. P.; Miller, J. J. H.; Schilders, W. H. A.: Uniform numerical methods for problems with initial and boundary layers, (1980) · Zbl 0459.65058
[17] Vanveldhuizen, M.: High order schemes of positive type for singular perturbation problems, Numerical analysis of singular perturbation problems, 361-383 (1979) · Zbl 0429.65087
[18] Kadalbajoo, M. K.; Patidar, K. C.: Variable mesh spline in compression for the numerical solution of singular perturbation problems, Int. J. Comput. math. 80, No. 50, 83-93 (2003) · Zbl 1017.65066 · doi:10.1080/00207160304656
[19] Chin, R. C.; Krasny, R.: A hybrid asymptotic finite element method for stiff two-point boundary value problems, SIAM J. Sci. stat. Comput. 4, No. 2, 229-243 (1983) · Zbl 0567.65055 · doi:10.1137/0904018