Images Maple Mathematica Matlab PowerPoint

Week 4

6/14 Week 1:

Rankine, Hugonoit, Napier, Burger

We study the formation and evolution of shockwaves from initial data given nonlinear hyperbolic systems of partial differential equations. Our emphasis is on the Riemann problem. Our introduction to the subject matter came in the form of Randall Leveque's book, Numerical Methods for Conservation Laws. Leveque writes these notes for an audience of first year graduate students or sufficiently prepared undergraduates. He derives the mathematical statement of a conservation law using integrals. He introduces the reader to the work of Rankine, Hugoniot, Euler, Napier and Burger.

The physical system our differential equation describes is the “Euler Shock tube”. It is a long thin pipe such that each slice of the pipe has the same value of the state variable throughout.

Different problems in PDEs have different names. “The Cauchy Problem” is to find how a state variable evolves given any function of space only that defines the state variable profile at time t=0. The evolution will be different depending on what system of differential equations is chosen. This problem is very hard to do algebraically and other mathematicians have used “the method of finite differences” to find approximate answers.

“The Riemann Problem” is to resolve a discontinuity in the state variable profile. Imagine that in our shocktube there is a barrier that prevents material from crossing it. On one side of the barrier it is very dense and the state variables take on one particular set of values and on the other side of the barrier it is not as dense and the state variables take on another particular set of values. The Riemann problem is to model what happens in a contact situation where the barrier is removed.

The “Rankine Hugoniot” jump conditions allow us to resolve this question. Discontinuities can be resolved in two ways. Either we insert a rarefaction or we insert a shock. Let us assume that we insert a shock. Through a series of manipulations starting with the integral conservation laws, it is commonly known that s[f]=[u] where s is the wavespeed, [f] is the jump in the flux function and [u] is the jump in the state variable. Solving for s we find the propagation speed of the discontinuity.

“Burger’s equation” is a refinement of our original equation where we take the flux function to be U^2. So our conservation form becomes, Ut+(U^2)x = 0. Our quasilinear form becomes Ut+U Ux = 0. .

Produced and maintained by Robert Chase, the Department of Mathematics and Statistics