High order finite-difference methods for two-point boundary value problems with singular sources (Q1079929)

The author constructs compact finite-difference schemes of arbitrary high order for the following problem with a single point source: \(L(u)=- u''+a(x)u'+b(x)u=\delta (x-y)\), \(0<x<1\); \(u(0)=u(1)=0\), where y is a fixed point \((0<y<1)\), \(\delta\) (.) is the Dirac delta function and a(.), b(.) are smooth and such that \(L(u)=0\), \(u(0)=u(1)=0\) imply \(u=0\). A complete discretization error analysis is given and numerical experiments confirming the convergence rates are presented.