Mono-Implicit Runge-Kutta-Nystr\"om Methods  
                  for 
Boundary Value Ordinary Differential Equations

         P.H. Muir and M. Adams


                Abstract

The successful use of mono-implicit Runge-Kutta methods
has been demonstrated by several researchers who have 
employed these methods in software packages for the 
numerical solution of boundary value ordinary differential
equations. However, these methods are only applicable to 
first order systems of equations while many boundary value 
systems involve higher order equations. While it is 
straightforward to convert such systems to first order, 
several advantages, including substantial gains in 
efficiency, higher continuity of the approximate solution 
and lower storage requirements, are realized when the 
equations can be treated in their original higher order 
form. In this paper, we consider generalizations of 
mono-implicit Runge-Kutta methods, called mono-implicit 
Runge-Kutta-Nystr\"om methods, suitable for systems of 
second order ordinary differential equations, having the 
general form, $y''(t)=f(t,y(t),y'(t))$, and derive optimal 
symmetric methods of orders two, four, and six. We also 
introduce continuous mono-implicit Runge-Kutta-Nystr\"om 
methods which yield interpolants for the discrete methods.
Numerical results are included to demonstrate the 
effectiveness of these methods. Savings of more than 70\% 
are attained in some instances.