Modern Infinitesimals and Delta-Function Perturbations of a Contact Discontinuity

Abstract

This article applies nonstandard analysis — a modern theory of infinitesimals — to study generalized-function perturbations of contact discontinuities in compressible, inviscid fluids. Nonstandard analysis is an area of modern mathematics that studies extensions of the real number system to nonstandard number systems that contain infinitely large and infinitely small numbers. Perturbations of a contact discontinuity are considered that represent one-dimensional analogs of the two-dimensional perturbations observed in the initial evolution of a Richtmyer-Meshkov instability on a density interface. Nonstandard predistributions of the Dirac delta function, d, and its derivatives, δ⁽ⁿ⁾, are applied as the perturbations of a contact discontinuity. The one-dimensional Euler equations are used to model the flow field of a fluid containing a perturbed density interface and generalized solutions are constructed for the perturbed flow field.