Convergence acceleration for boundary value problems with singularities using the E-algorithm

In the present work we use the E-algorithm to accelerate the convergence of finite-difference schemes for ordinary differential equations. As a model, a boundary-value problem for a second-order differential equation is considered, where the right-hand side may have different kinds of singularities. An asymptotic error expansion is obtained, enabling the use of the E-algorithm to accelerate the convergence. The applicability and the efficiency of the E-algorithm are discussed and illustrated by numerical examples.