SLIDE 1
Hyperbolicity singularities in rarefaction waves
Alexei Mailybaev
Moscow State University, Russia
Dan Marchesin
IMPA, Brazil
SLIDE 2 Outline
Structure of rarefaction waves near elliptic boundary
- regular points of elliptic boundary
- exceptional points of elliptic boundary
- classification and analysis of singularities
near exceptional points Exceptional points in Riemann problem for 2 conservation laws. Novel types of Riemann solutions:
- generic 1-wave solutions
- extreme non-uniqueness: infinite number of
unstable solutions for the same initial conditions Example
SLIDE 3 Rarefaction waves in systems of conservation laws
) ( =
U F t U
n equations in one space dimension x state vector U Rn, flux function F Simple wave solution
t x U t x U / ), ( ~ ) , ( = =
U F U A U U U A
( , ) (
r dU r r U A || , ) (
Real eigenvalues (characteristic speeds)
n
< < K
2 1
SLIDE 4
Elliptic boundary
Boundary between elliptic region (complex ) and hyperbolic region (real ) is given by double (in general, multiple) real eigenvalues 1 = 2 with a 22 Jordan block Regular point: eigenvector r is transversal to elliptic boundary Exceptional point: eigenvector r is tangent to elliptic boundary
SLIDE 5 Fold structure at regular points
Surface smooth parametrization
) ( , 4 )) ( ) ( ( ) ( , 2 ) ( ) ( ) (
* 2 1 2 1 2
U U U U p U U U s
=
) ( ), (
2
U s U p = =
( ) ( ,
1
U R U R r
= + + =
SLIDE 6 Rarefaction wave structure near regular points
Method: analysis is carried out in the extended (,,U)-space Jordan block structure at U*
, 1 , , , ,
1 1 1 1 1 1 1
= = + = = + = = r l r l l l A l l A l r r r A r r A
T T T T T T T
U r F l p
T
,
1 , 1 , 2
Rarefaction wave
) | (| ) ( ) (
2 2 *
=
p U U
SLIDE 7
Rarefaction wave structure near exceptional points: singularities in fold coordinates (,)
SLIDE 8
Rarefaction wave structure near exceptional points: singularities in state space
SLIDE 9 Quantitative description of singularities
Eigenvalues characterizing singularity type
( ) ( )
F l F l F l D D F l
T T T T 3 2 2 1 1 1
8 2 3 , 2 1
±
±
- Rarefaction wave curves passing through the exceptional point
F l F l s
s U U
T T 2 1 1 *
2 ), ( 2 ) ( 2 ) (
=
±
- Equation for inflection locus (points of rarefaction curves, where
characteristic speed attains maximum or minimum)
2 1 1
2 2
R p R s R p R s p p
SLIDE 10
Rarefaction waves in (x,t) space
Both cases are possible: family of a simple wave can increase or decrease when passing the exceptional point
SLIDE 11
Exceptional points for more than two conservation laws
Geometry of singularity in state space is determined by the eigenvector r0 and the associated vector (generalized eigenvector) r1
SLIDE 12 Riemann problem
Riemann problem initial conditions
< = = , , , , ) , ( x U x U t x U
R L
Solution is a sequence of shock waves (discontinuities) and rarefaction waves separated by constant states. Classical solution contains n waves, one for each characteristic family Shock waves: left state U, right state U+, speed s
) ( ) ( ) (
U s U F U F
Rankine-Hugoniot conditions Lax conditions
) ( ), ( ) ( : ) ( ), ( ) ( :
1 2 2 2 2 1 1 1
+
> < < < < < U s U s U S U s U s U S
SLIDE 13
Riemann solutions near regular points of elliptic boundary
Given left (right) state, rarefaction curves passing through regular points of elliptic boundary serve as bifurcation boundaries for Riemann solutions with different right (left) initial states.
SLIDE 14
Riemann solutions near exceptional points
A) Unique 3-waves solution (stable) B) Unique 1-wave solution (stable) C) Infinite number of 3-waves solutions (unstable) + one separate 2-waves solution (stable)
SLIDE 15 Example
Flux function
, 10 2 / 2 / ) (
2 1 2 2 2 1 2
+ = U U U U U U F
14 =
- Exceptional point: U* = 0
Singularity types: (b) < 0, (c) 0 < < 10 or 20 < , (d) 10 < < 20
SLIDE 16 Conclusion
Structure of rarefaction curves near regular and exceptional points of elliptic boundary is analyzed both qualitatively and quantitatively Novel types of Riemann solutions for 2 conservation laws containing exceptional points are found:
- generic stable 1-wave solutions
- extreme non-uniqueness: infinite number of
unstable solutions for the same initial conditions
