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On numerical solution of a turning point problem. (English) Zbl 0718.65055
This paper proposes finite difference schemes and non-uniform grids for the numerical solution of the singular perturbation problem: $$\epsilon^ 2u''+xb(u)u'=f(x),$$ $$x\in [a,1]$$, $$u(a)=A$$, $$u(1)=B$$ with $$a=0$$ or $$a=- 1$$. The author defines non-uniform meshes generated by a suitable function $$\lambda (x_ i=\lambda (ih)$$, $$i=1,...,n$$, $$h=1/n)$$ which is essentially a modification of the inverse of the boundary layer function. Then a finite-difference discretization is proposed and its stability in the discrete $$L_ 1$$-norm is proved under suitable assumptions. Finally, first order uniform convergence is proved for some class of $$\lambda$$- functions.
Reviewer: M.Calvo (Zaragoza)

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L12 Finite difference and finite volume methods for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34E15 Singular perturbations, general theory for ordinary differential equations