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. .