LAUSR

The purpose of this paper is to obtain the error bounds of fully geometric mesh one-leg methods for solving the nonlinear neutral functional differential equation with a vanishing delay. For this purpose, we consider Gq-algebraically stable one-leg methods which include the midpoint rule as a special case. The error of the first-step integration implemented by the midpoint rule on [0,T0] is first estimated. The optimal convergence orders of the fully geometric mesh one-leg methods with respect to T0 and the mesh diameter hmax$h_{\max }$ are then analyzed and provided for such equation. Numerical studies reported for several test cases confirm our theoretical results and illustrate the effectiveness of the proposed method.