Efficiency of methods for second-order problems

We discuss the efficient implementation of some fourth-order P-stable methods for solving implicit second-order ordinary differential equation initial-value problems. The methods are a two-stage implicit Runge-Kutta method and the class of direct hybrid methods introduced independently by Cash (1981) and Chawia (1981). We use techniques based on both complex and real arithmetic and appropriate predictors and local error estimators in the implementation. Algorithms based on the (second-order) trapezium rule are presented for comparison purposes. The results of some simple numerical tests are given. These tests allow a qualitative comparison of the proposed algorithms. © 1990 Oxford University Press.