Stiff Equation - Motivating Example

Motivating Example

Consider the initial value problem

The exact solution (shown in cyan) is

with as

We seek a numerical solution that exhibits the same behavior.

The figure (right) illustrates the numerical issues for various numerical integrators applied on the equation.

1. Euler's method with a step size of h = 1/4 oscillates wildly and quickly exits the range of the graph (shown in red).
2. Euler's method with half the step size, h = 1/8, produces a solution within the graph boundaries, but oscillates about zero (shown in green).
3. The trapezoidal method (i.e., the two-stage Adams–Moulton method) is given by
   Applying this method instead of Euler's method gives a much better result (blue). the numerical results decrease monotonically to zero, just as the exact solution does.

One of the most prominent examples of the stiff ODEs is a system that describes the chemical reaction of Robertson:

If one treats this system on a short interval, e.g. there is no problem in numerical integration. However, if the interval is very large (1011 say), then many standard codes fail to integrate it correctly.

Additional examples are the sets of ODEs resulting from the temporal integration of large chemical reaction mechanisms. Here, the stiffness arises from the coexistence of very slow and very fast reactions. To solve them, the software packages KPP and Autochem can be used.