Abstract
In this paper a nonlinear singularly perturbed initial problem is considered. The behavior of the exact solution and its derivatives is analyzed, and this leads to the construction of a Shishkin-type mesh. On this mesh a hybrid difference scheme is proposed, which is a combination of the second order difference schemes on the fine mesh and the midpoint upwind scheme on the coarse mesh. It is proved that the scheme is almost second-order convergent, in the discrete maximum norm, independently of singular perturbation parameter. Numerical experiment supports these theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amiraliyev, G.M., Duru, H.: A uniformly convergent finite difference method for a singularly perturbed initial value problem. Appl. Math. Mech. 20(4), 379–387 (1999)
Amiraliyeva, I.G., Erdogan, F., Amiraliyev, G.M.: A uniform numerical method for dealing with a singularly perturbed delay initial value problem. Appl. Math. Lett. 23(10), 1221–1225 (2010)
Cen, Z.: A second-order difference scheme for a parameterized singular perturbation problem. J. Comput. Appl. Math. 221(1), 174–182 (2008)
Cen, Z., Xu, A., Le, A.: A second-order hybrid finite difference scheme for a system of singularly perturbed initial value problems. J. Comput. Appl. Math. 234(12), 3445–3457 (2010)
Erdogan, F., Amiraliyev, G.M.: Fitted finite difference method for singularly perturbed delay differential equations. Numer. Algo. 59(1), 131–145 (2012)
Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust computational techniques for boundary layers. Chapman and Hall/CRC Press, New York (2000)
Gronwall, T.H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20(4), 292–296 (1918)
Kadalbajoo, M.K., Gupta, V.: A brief survey on numerical methods for solving singularly perturbed problems. Appl. Math. Comput. 217(8), 3641–3716 (2010)
Linß, T.: Uniform second-order pointwise convergence of a finite difference discetization for a quasilinear problem. Comput. Math. Math. Phys. 41(6), 898–909 (2001)
Linß, T.: Sufficient conditions for uniform convergence on layer-adapted grids. Appl. Numer. Math. 37(1-2), 241–255 (2001)
Linß, T.: Layer-adapted meshes for reaction-convection-diffusion problems. Berlin: Springer-Verlag (2010)
Lorenz, J.: Stability and monotonicity properties of stiff quasilinear boundary problems. Univ. u Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 12(1982)151–175.
Roos, H.-G., Stynes, M., Tobiska, L.: Numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems, 2nd edn. Springer-Verlag, Dordrecht (2008)
Stynes, M., Roos, H.-G.: The midpoint upwind scheme. Appl. Numer. Math. 23(3), 361–374 (1997)
Stynes, M., Tobiska, L.: A finite difference analysis of a streamline diffusion method on a Shishkin mesh. Numer. Algo. 18(3-4), 337–360 (1998)
Willett, D., Wong, J.S.W.: On the discrete analogues of some generalizations of Gronwall’s inequality. Monatshefte fr Mathematik. 69(4)362–367 (1965)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cen, Z., Erdogan, F. & Xu, A. An almost second order uniformly convergent scheme for a singularly perturbed initial value problem. Numer Algor 67, 457–476 (2014). https://doi.org/10.1007/s11075-013-9801-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-013-9801-0