On the construction of symmetric second order methods for ODEs

Abstract

We show how to build explicit symmetric second order methods for solving ordinary differential equations. These methods are very useful when low accuracy is required or when higher order ones by extrapolation or composition are desired to reach high accuracy. The proposed schemes are obtained by using simple splitting methods on an extended phase space. By construction, the schemes are symmetric and of second order allowing to recover most well known and frequently used schemes from the literature. This provides a simple proof on their time symmetric structure that is very useful when the schemes are used to get higher order methods by extrapolation or composition. We show how to obtain them in the general case as well as how to get Nyström methods, methods for stiff problems, Lie group integrators or symplectic integrators, but the technique can also be used to build explicit and implicit methods for many other problems.