Back to Departmental Colloquium: Fall 2008
Departmental Colloquium
Date: Thursday, Dec 11, 2008
Time: 4:15PM
Location: JWB 335
Roy Baty
Los Alomos
Title |
Nonstandard analysis and jump conditions for shock waves |
Abstract |
Nonstandard analysis is a relatively new area of mathematics in which infinitesimal numbers can be defined and manipulated rigorously like real numbers. To demonstrate the power of the subject, the problem of shock wave jump conditions is studied for a one-dimensional compressible gas. It is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval. To use conservation laws, pre-distributions of the Dirac delta measure are applied whose supports are contained within the shock thickness. Furthermore, piecewise differentiable pre-distributions of the Heaviside function are applied which vary from zero to one across the shock wave. It is shown that if the equations of motion are expressed in non-conservative form then the relationships between the jump functions for the flow parameters may be found unambiguously. The analysis yields the classical Rankine-Hugoniot jump conditions for an inviscid shock wave. Examples developed include a normal shock wave and a radially symmetric shock wave. |
Data Science and Machine Learning
Applied Mathematics