Differential Equation Basics, Andrew Witkin and David Baraff of Pixar Animation Studios, SIGGRAPH 2003 Course Notes.

Summary:

This paper discusses the basic strategy of explicit integration and the specific cases of Euler's method and the midpoint method. The paper also touches on how to calculate error in order to estimate adaptive step sizes and some issues to consider when implemeting an ODE solver. 

Problem:

How does one move a simulation forward simply? 
How can one avoid excessive errors when moving the simulation forward?

Method:

The basic method used for both Euler's method as well as the midpoint method is to take a set of descrete steps forward in time based on the current state and forces. Since these steps are trying to aproximate an underlying real valued function, some error is introduced with each step. The larger the step the greater the error. Euler's method essentially only models the first two terms in the Taylor series that would aproximate the actual function. Because of this, Euler's method accumulates error very quickly unless a small time step is used. A slightly better aproximation can improve the error problem. This leads to the midpoint method, which models the first three terms of the Taylor series and so will remain closer to the underlying function. 

Key Ideas:

	1. Explicit integration involves simple descrete steps and will accumulate significant error if the step size is too small to allow a good model of the underlying function.
	2. Euler's method and the midpoint method both perform explicit integration, but the midpoint method offeres a slight improvement in the amount of error introduced. 
	3. Addaptive step sizes can help to keep explicit integration from degenerating into chaos due to errors.

Contributions:

This paper is more of an explanatory, learning aid. It presents basic explicit integration in two forms (Euler's method and the midpoint method) in a very strait forward form and includes practical notes about error accumulation and adaptive stepsizes.

Questions:

While this paper did talk about how to estimate error for your adaptive step size, it wasn't really explicit about how you chose a good adaptive step size.