A second order accurate, positive scheme for singularly perturbed boundary value problems
A new finite difference scheme of positive type is proposed for the numerical solution of singularly perturbed two-point boundary value problems of type: \(Lu=-\epsilon u''+f(x)u'+g(x)u=q(y)\), \(u(0)=a\), \(u(1)=b\), where \(f,g>0\), q are smooth functions and the small parameter \(\epsilon \ll O(h)\). The finite difference method given here may be obtained applying a standard central difference scheme to the following perturbed version of \((1)\quad L_ 0w=-\epsilon_ 0(x)w''+f_ 0(x)w'+g_ 0(x)w=q_ 0(x),\) where \(\epsilon_ 0(x)=\epsilon_ 0+\delta f^ 2\), \(f_ 0=f-\delta ff'-\delta fg,\) \(g_ 0=g-\delta fg',\) \(q_ 0=q-\delta fq',\) and \(\delta\) is a suitable function. It is proved that under suitable assumptions on \(\delta\) the global errors are of order \(O(h^ 2+\epsilon h)\) in regions which exclude a few mesh points in the neighborhood of possible layers. Some numerical experiments with several test problems proposed by \textit{C. E. Pearson} [J. Math. Phys. 47, 134-154 (1968; Zbl 0167.158)] are considered, but it is not clear how to choose \(\delta\) in an optimal way. Finally an extension of this scheme for 2-D problems is presented.
finite difference scheme of positive type
singularly perturbed two-point boundary value problems
scientific article; zbMATH DE number 4030873