An efficient hybrid numerical scheme for convection-dominated boundary-value problems

This article presents a numerical scheme for convection-dominated two-point boundary-value problems. The proposed scheme combines the cubic spline scheme and the midpoint scheme in an appropriate manner. In the inner region, the convective term is approximated by three-point differences by spline approximation of solution at the mesh points, whereas in the outer region the midpoint approximations are used for convective term, and the classical central difference scheme is used for the diffusive term. The first-order derivative in the left boundary point is approximated by the cubic spline. This scheme is applied on the boundary layer resolving Shishkin mesh. Truncation error is derived, and the proposed method is applied to couple of examples to show its accuracy and efficiency.