Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals

Abstract

The classical numerical treatment of boundary value problems defined on infinite intervals is to replace the boundary conditions at infinity by suitable boundary conditions at a finite point, the so-called truncated boundary. A truncated boundary allowing for a satisfactory accuracy of the numerical solution has to be determined by trial and errors and this seems to be the weakest point of the classical approach. On the other hand, the free boundary approach overcomes the need for a priori definition of the truncated boundary. In fact, in a free boundary formulation the unknown free boundary can be identified with a truncated boundary and being unknown it has to be found as part of the solution.In this paper we consider a different way to overcome the introduction of a truncated boundary, namely non-standard finite difference schemes defined on quasi-uniform grids. A quasi-uniform grid allows us to describe the infinite domain by a finite number of intervals. The last node of such grid is placed on infinity so that right boundary conditions are taken into account exactly. We apply the proposed approach to the Falkner–Skan model and to a problem of interest in foundation engineering. The obtained numerical results are found in good agreement with those available in literature. Moreover, we provide a simple way to improve the accuracy of the numerical results using Richardson’s extrapolation. Finally, we indicate a possible way to extend the proposed approach to boundary value problems defined on the whole real line.