Numerical methods


Rounding errors
In general it is difficult to get a good analysis of rounding errors due to floating point arithmetic done by computers. To motivate the study of rounding errors we note that several disasters are originated by rounding errors. Recent examples are: Patriot Missile Failure and the Explosion of the Ariane 5. Rounding errors can also has important political and/or economical consequences. Examples are the index of The Vancouver Stock Exchange , the conversion to Euro's, and Rounding error changes Parliament makeup.

Numerical errors
The sinking of the Sleipner A offshore platform has been caused by inaccurate finite element approximation of the linear elastic model of the tricell (using the popular finite element program NASTRAN). The shear stresses were underestimated by 47%, leading to insufficient design.

Double pendulum
As an application of the theory given in Chapter 5 of this course a simulation (using a Java-applet) of a double pendulum is possible. The integration is done by a Runge-Kutta method.

Heat equation
In Chapter 6 of this course the numerical methods are applied to approximate the solution of the heat equation. There are many practical examples, which can be described by the heat equation. One of them is the diffusion of oxygen in the skin. In some radio therapies the best results are obtained if the oxygen concentration is as low as possible. The outer surface of the skin is sealed off and the concentration decreases, due to the fact that the skin uses the oxygen at a certain rate. For different times the concentration profiles are calculated and given in the figure below:

Here is an animation of the concentration profiles

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