Geometric Foundations of Numerical Algorithms and Symmetry

This paper outlines a new general construction, named \multi-space", that forms the proper geometrical foundation for the numerical analysis of dierential equations | in direct analogy with the role played by jet space as the basic object underlying the geometry of dierential equations. Application of the theory of moving frames leads to a general framework for constructing symmetry-preserving numerical approximations to dierential invariants and invariant dierential equations