Explicit Ordinary Differential Equations

Most ordinary differential equations (ODEs) cannot be solved analytically. Only for a restricted class of ODEs computing the analytical solution can be taken into consideration. But even in these cases it is often more efficient to apply numerical techniques to approximate the solution. 

Numerical integration is the backbone of all simulation tools. Understanding the basics of these methods can help a lot in interpreting the results. Bad or unexpected performance of such a method can be a hint to modeling errors and can motivate either the choice of another method or a modification of the model at hand. 

In this chapter the basics of numerical integration methods for explicit ODEs will be presented. The goal is to discuss the fundamental ideas, eg. order, stability, step size selection etc.. We will focus also on the main components of modern implementations of ODE solvers and point out special modifications which can exploit the structure of equations of motion of mechanical systems. 

models are considered to be simple , if it is an easy task to set up their equations of motion, or if there are few equations which perhaps are nearly linear. Being simple in that sense does not necessarily imply that the integration task is easy. We will see this when discussing stiff ODEs. 

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This requires the numerical solution of explicit ordinary differential equations or differential-algebraic equations, see Chs. 4, 5, 6. 

In the last chapter we discussed the numerical treatment of explicit ordinary differential equations. Here, we will consider the more general case, implicit ordinary differential equations. 

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