Numerical Treatment of Differential Equations by R. Bulirsch, R.D. Grigorieff, J. Schröder

By R. Bulirsch, R.D. Grigorieff, J. Schröder

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Still another example of this kind of parallel and independent stimuli is furnished by the consideration of the classical Riemann spaces in the general theory of relativity. Here, too, the mathematical tools were there for Einstein to use for his formulation of the geometry of space. Einstein's theory of special relativity initiated a revolutionary departure from the restrictions on the scope of geometrical ideas by involving, in addition to the physical "space," the variable or the parameter of time.

The element is a set of points and the distance between two sets can be defined in a simple way as described by the mathematician Hausdorff. Given a point x of set B, look at the nearest point y in set A and then take the maximum of the minima with respect to all choices of x. That will be the distance between sets. We wrote a paper, the first I ever published in this country. We sent it to the Bulletin of the American Mathematical Society in 1932. In it we showed that for two dimensions you get a space that you might imagine just as a triangle, for three dimensions you can imagine it as a sort of tetrahedron, but topologically they are the same as a square or a cube.

The first meaningful applications were made in problems of mathematical physics, where the complexity of the interaction of many elements may be such that no methods of classical mathematical analysis can give even a qualitative picture of their solution. In hydrodynamics (the study of the motion of fluids) one must have recourse (other than in very simple cases) to numerical work which becomes so massive that only the fastest machines can provide useful approximations. In problems of weather prediction or of the circulation of the atmosphere, initial data also lacks the simplicity and symmetry that allow for the use of theoretical methods of analysis.

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