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Page 1: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

SIAM Conference on Analysis of PDEs (PD11),

November 14-17, 2011,

San Diego Marriott Mission Valley, San Diego

MS74: Multidimensional Conservation Laws and

Related Applications - Part III

Riemann Problems for Two

Dimensional Euler Systems

Yuxi Zheng

Yeshiva University

Page 2: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Euler Equation

ρt +∇ · (ρu) = 0,

(ρu)t +∇ · (ρu⊗ u+ pI) = 0,

(ρE)t +∇ · (ρEu+ pu) = 0.

(1)

Cauchy problems: Open

Page 3: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

We look for self-similar solutions

(u, v, p, ρ)(ξ, η)

for

ξ =x

t, η =

y

t.

Page 4: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

The Euler system becomes the self-similar form

−ξρξ − ηρη + (ρu)ξ + (ρv)η = 0

−ξ(ρu)ξ − η(ρu)η + (ρu2 + p)ξ + (ρuv)η = 0

−ξ(ρv)ξ − η(ρv)η + (ρuv)ξ + (ρv2 + p)η = 0

−ξ(ρE)ξ − η(ρE)η + (ρuE + pu)ξ + (ρvE + pv)η = 0.

(2)

Page 5: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Any initial data has to be

(u, v, p, ρ)(0, x, y) = Independent of radius

The so-called Riemann problem.

It is a boundary-value problem: Initial data be-

comes boundary data at inifinity.

Page 6: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Riemann problems: 2-D

Figure 1: Four−constant Riemann problem

constant state 3

constant state 1

constant state 4

x

y

constant state 2

Page 7: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

State 2

State 1

The characteristics are tangent to the sonic curve.of four forward rarefaction waves in the 1990 paper.

η

P

Qsu

bson

ic

a

����

ξ

Figure 2. Proposed structure for the interaction

Z

��������

����

Page 8: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Numerical simulations:

1. Tong Zhang, Guiqiang Chen, Shuli Yang;

2. Lax Liu;

3. Schulz-Rinne, Collins, and Glaz;

4. Tadmore , Kurganov;

5. Glimm, J., Xiaomei Ji; Jiequan Li, Xiaolin Li;

Peng Zhang; Tong Zhang; Yuxi Zheng;

6. W.C. Sheng et al

Page 9: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Con

figur

atio

n 1

Density T=0.2

Figure 3: Density contour lines, courtesy of Lax and Liu.

Page 10: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

4R (Two forward, two backward)

Page 11: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

+1.0

Page 12: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

First, two wave interaction:

Hodograph method: paper with Jiequan Li, ARMA

(vol193(2009)).

Direct method (with Xiao Chen)(IUMJ, 59(2010))

with Jiequan Li and Zhicheng Yang, (JDE (2010)).

Then: Simple wave (With Tong Zhang and Jiequan

Li: CMP, v.267(2006))

Page 13: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Then: Semi-hyperbolic patches

Page 14: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

+1.0

Page 15: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Semi-hyperbolic patch in channel flow

Page 16: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

1. With Kyungwoo Song, on pressure gradient

system, DCDS (2009)

2. With Mingjie Li (Ph.D student from CNU)

ARMA 2011

Page 17: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Recently, in Tianyou Zhang’s thesis, we estab-

lish regularity via a new existence approach for

the pressure gradient system

ut + px = 0,vt + py = 0,pt + pux + pvy = 0,

Or

(pt

p)t − pxx − pyy = 0.

Page 18: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

In the self- similar plane

(p− ξ2)pξξ − 2ξηpξη + (p− η2)pηη

+1p(ξpξ + ηpη)2 − 2(ξpξ + ηpη) = 0.

In polar coordinates,

(p− r2)prr +p

r2pθθ +

p

rpr +

1

p(rpr)

2 − 2rpr = 0.

Sonic line.: r2 − p = 0;

Hyperbolic where r2 − p > 0; Elliptic in the

region r2 − p < 0.

Page 19: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

D’Alembert type decomposition:

∂+∂−p = q(∂+p− ∂−p)∂−p,∂−∂+p = q(∂−p− ∂+p)∂+p,

where

∂± = ∂θ ± λ−1∂r

λ =

p

r2(r2 − p)

q =r2

4p(r2 − p).

Page 20: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Introducing R = ∂+p, S = ∂−p,W = (p, R, S),

pθ = R+S2 ,

Rθ − λ−1Rr = q(S −R)R,

Sθ + λ−1Sr = q(R− S)S.

Page 21: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

For the semi-hyperbolic patch solution constructed

in Song and Zheng, pr may blow up. We do not

expect better regularity. In numerical simulations,

demarcation line between hyperbolic region and

elliptic region is composed of the sonic line and

a shock curve. Although the sonic curve changes

dramatically close to the shock, for a large portion

away from connection point with shock it is quite

smooth and pressure p changes smoothly across

it. We mainly focus on this part to propose our

problem for today’s talk.

Page 22: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Problem 1: Given a sonic curve r = f(θ), θa <

θ < θb, with

f ′ ≥ γ > 0,

denoted by Γ. We look for solutions to the PGE in

a class of functions with given first-order deriva-

tive pθ(f−1(r), r) = a0(r)

2 on Γ. We assume

a20 ≥ α > 0.

As a result pr = 2f − pθf ′ ≡ a1(r) is also known on

the sonic curve. We assume pθθ is bounded at the

sonic line.

Page 23: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

We show that the precise boundary conditions

are

W |Γ = (r2, a20, a20),

(pθ,√

r2 − pRθ,√

r2 − pSθ)|Γ = (a20,−a20a12 ,

a20a12 ).

Using the hyperbolicity quantity t =√

r2 − p and

r as new independent variables, and introducing

new dependent variables

U(t, r) = R(t, r)− a20(r)− a1(r)t,

V (t, r) = S(t, r)− a20(r) + a1(r)t,

Page 24: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

The system is transformed into

Ut +2tλ−1

S+2rλ−1Ur = b1(U, V ),

Vt − 2tλ−1

R−2rλ−1Vr = b2(U, V ),

where b1(U, V ) and b2(U, V ) contain a singular

term U−Vt and depend on a0 and a1.

b1(U, V ) = 12

(

U−Vt

)

+

(

2t2qRS+2rλ−1 − 1

2

)

(

U−Vt +2a1

)

−f(a0, a1, V ),

b2(U, V ) = 12

(

V−Ut

)

+

(

2t2qSR−2rλ−1 − 1

2

)

(

V−Ut − 2a1

)

+g(a0, a1, U),

Page 25: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

and

f(a0, a1, V ) = 2tλ−1

S+2rλ−1

(

∂r(a20 + ta1))

,

g(a0, a1, U) = 2tλ−1

R−2rλ−1

(

∂r(a20 − ta1))

.

The boundary conditions are

U(0, r) = V (0, r) = Ut(0, r) = Vt(0, r) = 0,

(ra ≤ r ≤ rb.)

By establishing an iteration scheme we establish

the existence of a solution by showing that itera-

tion sequence converges under a weighted metric

Page 26: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

through characteristic analysis. The main result

is stated as follows:

Theorem. Assume a0 and a1 are in C3([ra, rb])

and D(δ0) is a strong determinate domain, then

there exists a δ > 0 such that on D(δ), the degen-

erate hyperbolic problem has a classical solution

in S.

Proof.

Now λ−1(t, r) = tr√r2−t2

and q(t, r) = r2

4(r2−t2)t2.

Page 27: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Let

D(δ0) := {(t, r)|0 ≤ t ≤ δ0, r1(t) ≤ r ≤ r2(t)},

where r1(t), r2(t) are continuously differentiable

on 0 ≤ t ≤ δ0, r1(0) = ra, r2(0) = rb and r1 < r2for 0 ≤ t ≤ δ0.

Definition a strong determinate domain D(δ0).

Page 28: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Solutions of the Cauchy problem will be ob-

tained within the class S consisting of all continu-

ously differentiable functions F =

(

f1f2

)

: D(δ) →

R2 satisfying the following properties:

(S1) fi(0, r) = ∂tfi(0, r) = 0 (i = 1,2),

(S2)∥

f1t2

0+∥

f2t2

0≤ M ,

(S3)∥

∂rf1t2

0+∥

∂rf2t2

0≤ M ,

Page 29: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

(S4) Fr is Lipschtz continuous with respect to r

with

∂2rrf1t2

0+

∂2rrf2t2

0≤ M ,

where || · ||0 is the superem norm over domain

D(δ) and 0 < δ ≤ δ0. We will denote W the

larger class containing only continuous functions

on D(δ) which satisfy the first two conditions

(S1) and (S2). Both S and W are subsets of

C0(D(δ);R2).

Page 30: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

We define a new weighted metric on S and W :

d(F,G) :=

f1 − g1t2

0+

f2 − g2t2

0. (3)

We will construct an integration iteration.

Assume u(t, r) and v(t, r) are admissible func-

tions on D(δ), i.e. they belong to set S. Differ-

entiate along the two characteristics defined by:

d

d+(v):= ∂t +Λ+(v)∂r,

d

d−(u):= ∂t +Λ−(u)∂r.

(4)

Page 31: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

The iteration :

d

d+(v)U = b1(u, v),

d

d−(u)V = b2(u, v). (5)

A mapping is thus determined: T

((

uv

))

=(

UV

)

.

Important properties of mapping T are given in

the following lemmas.

Page 32: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Lemma 1. There exist positive constants δ, M

and 0 < β < 1 such that: T maps S into S and

for any pair F, F̂ in S, there holds d(

T(F), T(F̂))

≤β d

(

F, F̂)

. The constants depend only on the C3

norms of a0, a1, α and D(δ0).

For any F(1) ∈ S, let F(n) = TF(n−1), then

the iteration sequence{

F(n)}

is Cauchy in (W,d)

which is a complete metric space.

The completeness of (W, d) can be proved by

showing d is a metric and the Cauchy sequence

Page 33: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

in (W, d) is actually convergent to an element in

the same space; all these are elementary argu-

ments. However since (S, d) is not complete, the

limit is only guaranteed to be in W but might not

stay in S. The limit will become the solution if

it further has certain regularity say differentiable,

which comes out of equi-continuity of the itera-

tion sequence{

F(n)}

which is guaranteed by fol-

lowing lemmas.

Page 34: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Lemma 2. The iteration sequence{

F(n)}

has

property that{

∂rF(n)}

and{

∂tF(n)}

are uniformly

Lipschitz continuous on D(δ).

Page 35: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Proof is complete: A smooth solution at a sonic

curve exists.

Page 36: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Similar transformations are done for the steady

and self-similar Euler systems.

Page 37: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Self-similar Euler: Use

t =√

U2 + V 2 − c2/c, φ

where

φξ = U = u− ξ, φη = V = v − η

is the pseudo-potential.

For steady Euler: Use

t =√

q2 − c2, and φ

where φx = u, φy = v.

Page 38: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Steady Euler in (t, φ):

∂tR− t2µ2

c2S∂φR = − Rq2

2Sc2t{R+S+[2(m+1) cos4 ω−

(m+3)cos2 ω]S}

∂tS− t2µ2

c2R∂φS = − Sq2

2Rc2t{S+R+[2(m+1) cos4 ω−

(m+3)cos2 ω]R}

∂tc− 2t2µ2

c2(R+S)∂φc = −tµ2

c .

Here m = (3− γ)/(γ+1), ω = (α−β)/2, tanα =

Λ+, tanβ = Λ−, µ2 = (γ − 1)/(γ + 1), cos2 ω =

t2/q2, R = ∂̄+c/c, S = ∂̄−c/c.

Page 39: Riemann Problems for Two Dimensional Euler Systems...2011/11/16  · The so-called Riemann problem. It is a boundary-value problem: Initial data be-comes boundary data at inifinity.

Thank you.