# zbMATH — the first resource for mathematics

A uniform numerical method for quasilinear singular perturbation problems without turning points. (English) Zbl 0664.65082
Consider the singularly perturbed boundary value problem $$(1)\quad Tu:=- \epsilon u''-b(x,u)u'+c(x,u)=0,$$ $$x\in I=[0,1]$$, $$Bu:=(u(0),u(1))=(U_ 0,U_ 1),$$ where the functions b, c and the numbers $$U_ 0$$, $$U_ 1$$ are given, $$\epsilon$$ is a small parameter, $$0<\epsilon \leq \epsilon_ 0$$. Suppose that $$b,c\in C^ 2$$ (I$$\times R)$$, $$b(x,u)\geq \beta >0$$, $$b_ x(x,u)+c_ u(x,u)\geq \gamma$$, $$\gamma\leq 0$$, $$x\in I$$, $$\beta^ 2+4\epsilon \gamma >0$$. Then (T,B) is inverse monotone and there exists a unique solution $$u\in C^ 4(I)$$ of the problem (1). It has a boundary layer near the origin. To approximate u uniformly in $$\epsilon$$, the problem (1) is first transformed by introducing a special new independent variable. Then the Engquist-Osher type finite-difference scheme is applied. Stability uniform in $$\epsilon$$ is proved in the discrete $$L^ 1$$-norm. First order consistency uniform in $$\epsilon$$ and first order convergence uniform in $$\epsilon$$ in the discrete $$L^ 1$$-norm are proved. Numerical examples illustrate the pointwise covergence uniform in $$\epsilon$$.
Reviewer: L.Bakule

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34E15 Singular perturbations, general theory for ordinary differential equations
Full Text:
##### References:
 [1] Abrahamsson, L., Osher, S.: Monotone difference schemes for singular perturbation problems. SIAM J. Numer. Anal.19, 979–992 (1982). · Zbl 0507.65039 · doi:10.1137/0719071 [2] Berger, A. E., Solomon, J. M., Ciment, M.: An analysis of a uniformly accurate difference method for a singular perturbation problem. Math. Comput.37, 79–94 (1981). · Zbl 0471.65062 · doi:10.1090/S0025-5718-1981-0616361-0 [3] Doolan, E. P., Miller, J. J. H., Schilders, W. H. A.: Uniform Numerical Methods for Problems with initial and Boundary Layers. Dublin: Boole Press 1980. · Zbl 0459.65058 [4] Kellogg, R. B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comput.32, 1025–1039 (1978). · Zbl 0418.65040 · doi:10.1090/S0025-5718-1978-0483484-9 [5] Liseikin, V. D., Yanenko, N. N.: On the numerical solution of equations with interior and exterior boundary layers on a nonuniform mesh. In: BAIL III – Proceedings (Miller, J. J. H., ed.), pp. 68–80. Dublin: Boole Press 1984. · Zbl 0672.65070 [6] Lorenz, J.: Combinations of initial and boundary value methods for a class of singular perturbation problems. In: Numerical Analysis of Singular Perturbation Problems (Hemker, P. W., Miller, J. J. H., eds.), pp. 295–315. London: Academic Press 1979. · Zbl 0429.65086 [7] Lorenz, J.: Stability and monotonicity properties of stiff quasilinear boundary problems. Univ. u Novom Sadu Zb. rad. Prir.-Mat. Fak. Ser. Mat.12, 151–175 (1982). · Zbl 0546.34046 [8] Lorenz, J.: Numerical solution of a singular perturbation problem with turning points. Lecture Notes in Math.1017, 432–439 (1983). · Zbl 0517.65062 · doi:10.1007/BFb0103269 [9] Niijima, K.: On a difference scheme of exponential type for a nonlinear singular perturbation problem. Numer. Math.46, 521–539 (1985). · Zbl 0577.65070 · doi:10.1007/BF01389657 [10] O’Malley, R. E., Jr.: Introduction to Singular Perturbations. New York, London: Academic Press 1974. [11] Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. New York, London: Academic Press 1970. · Zbl 0241.65046 [12] Roos, H.-G.: A second order monotone upwind scheme. Computing36, 57–67 (1986). · Zbl 0572.65063 · doi:10.1007/BF02238192 [13] 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 [14] Vulanović, R.: On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh. Univ. u. Novom Sadu Zb. rad. Prir.-Mat. Fak. Ser. Mat.13, 187–201 (1983). · Zbl 0573.65064 [15] Vulanović, R., Herceg, D., Petrović, N.: On the extrapolation for a singularly perturbed boundary value problem. Computing36, 69–79 (1986). · Zbl 0576.34019 · doi:10.1007/BF02238193 [16] Vulanović, R.: A second order uniform method for singular perturbation problems without turning points. In: V. Conference on Applied Mathematics (Bohte, Z., ed.), pp. 183–194. Ljubljana: Univ. Ljubljana 1986. · Zbl 0606.65057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.