Runge-Kutta methods represent a popular trade-off between simplicity and efficiency. However, by focusing on the integration problem more carefully, it is often possible to improve efficiency further. The Euler and Runge-Kutta methods are often referred to as single-step methods. There exist multi-step methods, which rely on the fact that a sequence of integrations will be performed, in a manner analogous to incremental collision detection in Section 5.3.3. The key issues are ensuring that the methods properly initialize, ensuring numerical stability over time, and estimating error to adaptively adjust the step size. Many books on numerical analysis cover multi-step methods [51,440,863]. One of the most popular families is the Adams methods.
Multistep methods require more investment to understand and implement. For a particular application, the decision to pursue this route should be based on the relative costs of planning, collision detection, and numerical integration. If integration tends to dominate and efficiency is critical, then multi-step methods could improve running times dramatically over Runge-Kutta methods.
Steven M LaValle 2012-04-20