Article: Coordinate free numerics : Closing the gap between 'pure' and 'applied' mathematics?
Authors: H. Munthe-Kaas, M. Haveraaen
Journal: Zeitschrift für angewandte Mathematik und Mechanik, 1996, vol. 76, pp. 487-488
Abstract: The theory of differential equations have diverged in two different directions in our century; the applied mathematical coordinate based presentation which has its origin in hand calculations, and the pure mathematical presentation based on coordinate free formulations, commutative diagrams and global analysis. Through various examples, we show that much can be gained by employing 'abstract' coordinate free formulations also in scientific/numerical computing. The origin of this line of work, was an attempt to design a software system for numerical solution of tensor field equations on parallel computers by using formal methods from pure computer science [2]. It was discovered that the pure abstract mathematical definitions were better suited for software design than the concrete coordinate based definitions, and that modern computer languages enables us to express and program various numerical algorithms directly within a coordinate free language. Later [3] we discovered very useful computational techniques for manipulating tensor indexes which are simple to program within the abstract framework, but cannot be conveniently expressed within a coordinate based formulation. Recently [4], we have shown that the Butcher theory for analyzing Runge-Kutta methods may be replaced by a coordinate free analysis based on Lie series and Lie groups. Through its geometric formulation and structural simplicity, this approach is interesting for investigating geometric ODE integrators (e.g. symplectic ODE methods).
Hindsight notes: Coordinate free abstractions in computational mathematics was an important idea which lead to the investigation of numerical Lie group integrators.
1 Jul 1995