On the absolute stability of a two-step third order method on a graded mesh for an initial-value problem

Abstract

A two-step third order method on a variable mesh for the approximation of nonlinear IVP: $u^{\prime\prime}=f(t,u,u{^{\prime}}), u({t}_{0})={\gamma }_{0}$,$u{^{\prime}}({t}_{0})={\gamma }_{1}$ is proposed. For computation, only a monotonically decreasing mesh will be employed. The method when applied to a test equation $u^{\prime\prime}+2\alpha u{^{\prime}}+{\beta }^{2}u=g(t), \alpha >\beta \ge 0,$ is shown to be unconditionally stable. The proposed method is applicable to solve singular problems. A special technique is required to compute the method near the singular point. Several problems of physical significance including three problems on boundary layer are examined to illustrate the convergent character and usefulness of the approximation. Approximate solutions are provided to validate the functionality of the suggested approximation.