A third-order finite difference method on a quasi-variable mesh for nonlinear two point boundary value problems with Robin boundary conditions

Abstract

This article presents a third-order accurate finite difference approximation for a general nonlinear second-order ordinary differential equation subject to Robin boundary conditions. A quasi-variable mesh with three grid points is constructed by introducing a successive mesh ratio parameter $\eta$. For $\eta = 1$, the mesh turns uniform, and the scheme transforms into a fourth-order one. First, a third-order discretization for the given differential equation is derived at each internal grid point, and further, two fourth-order discretizations are designed at the boundary points. The convergence analysis of the proposed method is discussed in the entire solution domain including two boundary points. Since the scheme involves ‘off-step’ mesh points, it is directly applicable to singular problems, which is its primary advantage. Moreover, the flexibility in the choice of $\eta$ enables us to construct a denser grid in the region of boundary layer and a sparser one outside it; hence, the boundary layer problems are effectively dealt with. This is further demonstrated by successful application of the proposed technique over twelve problems of physical significance including singularly perturbed, singular, nonlinear viscous Burgers’ equations and boundary layer problems. Computational results validate the order of the proposed technique, and a comparison with the already existing results in the recent past reveals its superiority.