Hyperbolicity singularities in rarefaction waves Alexei Mailybaev - PowerPoint PPT Presentation

Hyperbolicity singularities in rarefaction waves Alexei Mailybaev Dan Marchesin Moscow State University, Russia IMPA, Brazil

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

Rarefaction waves in systems of conservation laws n equations in one space dimension x U F ( U ) � � 0 + = state vector U � R n , flux function F t x � � ~ U ( x , t ) U ( ), x / t Simple wave solution = � � = U U F � � � A ( U ) , A ( U ) Equation for U ( � ) = � = U � � � � � A ( U ) r r , dU || r Eigenvalue problem = � Real eigenvalues K � < � < < � (characteristic speeds) 1 2 n

Elliptic boundary Boundary between elliptic region (complex � ) and hyperbolic region (real � ) is given by double (in general, multiple) real eigenvalues � 1 = � 2 with a 2 � 2 Jordan block Regular point: eigenvector r is Exceptional point: eigenvector r is transversal to elliptic boundary tangent to elliptic boundary

Fold structure at regular points , r R ( U ) R ( U ) Surface smooth � = � + � + � = + � 0 0 1 parametrization 2 p ( U ), s ( U ) � = � = 2 ( U ) ( U ) ( ( U ) ( U )) � + � � � � 2 1 2 1 s ( U ) , p ( U ) , ( U ) = � � = � = � 0 0 * 2 4

Rarefaction wave structure near regular points A r r , A r r r , = � = � + Jordan block 0 0 0 0 0 1 0 1 0 structure at U * T T T T T T T l A l , l A l l , l r 1 , l r 0 = � = � + = = 0 0 0 0 1 0 0 1 0 0 1 1 1 Regularity condition T 2 p l F 0 , � = � � 0 0 0 r / U � = � � � 0 , 1 0 , 1 Rarefaction wave 2 ( ) � � � U ( ) U 0 r � = + * 0 p � 0 2 o (| | ) + � � � 0 Method: analysis is carried out in the extended ( � , � , U )-space

Rarefaction wave structure near exceptional points: singularities in fold coordinates ( � , � )

Rarefaction wave structure near exceptional points: singularities in state space

Quantitative description of singularities Eigenvalues characterizing singularity type 1 2 ( ) ( ) T T T 2 T 3 l F D , D 3 l F 2 l F 8 l F � = � � ± = � � � � + � 0 1 0 0 1 0 1 0 0 0 ± 2 Rarefaction wave curves passing through the exceptional point 2 ( ) � � � 0 U ( ) U r o ( ), � = + + � � � * 0 0 2 s � + � 0 ± T T 2 2 s l F l F � = � � + � 0 0 1 0 1 0 Equation for inflection locus (points of rarefaction curves, where characteristic speed � attains maximum or minimum) 2 2 p s R p R � � � � + � � 1 0 p � � = � � 2 s R p R � � + � � � � 0 1

Rarefaction waves in ( x , t ) space Both cases are possible: family of a simple wave can increase or decrease when passing the exceptional point

Exceptional points for more than two conservation laws Geometry of singularity in state space is determined by the eigenvector r 0 and the associated vector (generalized eigenvector) r 1

Riemann problem U , x 0 , < � Riemann problem L U ( x , t 0 ) = = � initial conditions U , x 0 , > � R 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 F ( U ) F ( U ) s ( U U ) � = � Rankine-Hugoniot conditions + � + � S : ( U ) s ( U ), s ( U ) � < < � < � 1 1 1 2 + � + Lax conditions S : ( U ) s ( U ), s ( U ) � < < � > � 2 2 2 1 + � �

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.

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)

Example 2 2 U U / 2 U / 2 � � + � + 2 1 2 F ( U ) , � is a parameter Flux function � � = � � 10 U U � � 1 2 Exceptional point: U * = 0 Singularity types: (b) � < 0, (c) 0 < � < 10 or 20 < � , (d) 10 < � < 20 14 � =

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