On the role of mathematical abstractions for scientific computing

Abstract

A distinguished feature of scientific computing is the necessity to design software abstractions for approximations. The approximations are themselves abstractions of mathematical models, and the mathematical models are also abstractions of the real world. In this paper, the relation between different mathematical abstraction levels and scientific computing software is discussed, in particular with respect to the simulation of partial differential equations. By applying software engineering practices already when the mathematical model is considered, a coordinate-free formulation is readily motivated. For partial differential equations, the continuos layer is identified as separate from the the coordinate-free layer. By mapping these layers into different software modules, numerical discretization carried out in the continuous layer is cleanly separated from the coordinate-free layer. This separation of concerns increases modularity because reusability is promoted. It is therefore concluded that the continuous and coordinate-free abstraction layers provide a solid foundation for software that simulates partial differential equations. © 2001 by Springer Science+Business Media New York.