Computable scalar fields: A basis for PDE software

Abstract

Partial differential equations (PDEs) are fundamental in the formulation of mathematical models of the physical world. Computer simulation of PDEs is an efficient and important tool in science and engineering. Implicit in this is the question of the computability of PDEs. In this context we present the notions of scalar and tensor fields, and discuss why these abstractions are useful for the practical formulation of solvers for PDEs. Given computable scalar fields, the operations on tensor fields will also be computable. As a consequence we get computable solvers for PDEs. The traditional numerical methods for achieving computability by various approximation techniques (e.g., finite difference, finite element or finite volume methods), all have artifacts in the form of numerical inaccuracies and various forms of noise in the solutions. We hope these observations will inspire the development of a theory for computable scalar fields, which either lets us understand why these artefacts are inherent, or provides us with better tools for constructing these basic building blocks.