Modern infinitesimals and the entropy jump across an inviscid shock wave

Abstract

This article applies nonstandard analysis to study the generalized solutions of entropy and energy across one-dimensional shock waves in a compressible, inviscid, perfect gas. Nonstandard analysis is an area of modern mathematics that studies number systems that contain both infinitely large and infinitely small numbers. For an inviscid shock wave, it is assumed that the shock thickness occurs on an infinitesimal interval and that the jump functions for the field variables are smoothly defined on this interval. A weak converse to the existence of the entropy peak is derived and discussed. Generalized solutions of the Euler equations for entropy and energy are then derived for both theoretical and realistic normalized velocity profiles.