Absolute stability of an implicit method based on third-order off-step discretization for the initial-value problem on a graded mesh

Abstract

We studied a new implicit method of order three based on two off-step points for the numerical solution of second-order nonlinear initial-value problems $y^{\prime\prime} = f(t,y,y^{\prime}),\;y(t_{0} ) = \gamma_{0}$, $y^{\prime}(t_{0} ) = \gamma_{1}$ on a graded mesh. In practice, only a monotonically decreasing mesh will be employed. We applied the proposed method to a test equation $y^{\prime\prime} + 2\alpha y^{\prime} + \beta^{2} y = g(t), \quad \alpha > \beta \ge 0,$ and analyse the stability which is absolutely stable for a graded mesh and superstable for a uniform mesh. The proposed method is applicable to solve singular problems. Seven benchmark problems including three boundary layer problems are solved to demonstrate the stability and utility of the method discussed. Numerical results are given to confirm the usefulness of the developed method.