HAL Id: hal-00546919https://hal.archives-ouvertes.fr/hal-00546919v1

Submitted on 15 Dec 2010 (v1), last revised 4 Mar 2014 (v2)

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Random sampling remap for compressible two-phaseflows

Mathieu Bachmann, Philippe Helluy, Jonathan Jung, Hélène Mathis, SiegfriedMüller

To cite this version:Mathieu Bachmann, Philippe Helluy, Jonathan Jung, Hélène Mathis, Siegfried Müller. Random sam-pling remap for compressible two-phase flows. Computers and Fluids, Elsevier, 2013, 86, pp.275-283.<10.1016/j.compfluid.2013.07.010>. <hal-00546919v1>

P P P

❲P ❲

P ❯❨ Ü

strt ♥ ts ♣♣r rss t ♣r♦♠ ♦ s♦♥ rt② sq ♦♠♣rss ♦s t♦t ♣rssr ♦st♦♥s t t sq ♥tr ❲ ♥tr♦ ♥ r♥♣r♦t♦♥ s♠ s ♦♥ r♥♦♠s♠♣♥ t♥q ♥tr♦ ② ♦♥s ♥ ♦t♥ ♥ ❬❪ ❲ ♦♠♣rt t♦ ♦st ♣♣r♦ ♥tr♦ ♥ ❬❲❪ ♥ ❬❪ s♣tt ♥♦♥♦♥srt tr ♦ t s♠s ♦sr t ♥♠r ♦♥r♥ t♦rs t r♥t s♦t♦♥ ♦r s♦♦♥tt ♥trt♦♥ tstss ♥② ♣♣② t ♥ s♠ t♦ t ♦♠♣tt♦♥ ♦ t ♦st♦♥s♦ s♣r r ♥s tr

♥tr♦t♦♥

♣rs♦♥ ♦ ♦♥srt ♦♥♦ s♠s ♣♣ t♦ t♦ ♦ss st tt s ♥ st ♥♦ ♦r ♠♦r t♥ t♥t② ②rs s ❬r ❪ ♥ ♥ rr♥s s ♣rs♦♥♠♣s ♣rtrt♦♥s ♦♥ t ♣rssr ♣r♦s tt r ♦t♥ t ♣rssr♦st♦♥s ♣♥♦♠♥♦♥

♦r t ♠♦♠♥t t s ♥♦t ♥ ♣♦ss t♦ s♥ s♠♣ ♦♥srt s♠tt ♣rsrs t ♦♥st♥t ♦t②♣rssr stts s ♣r♦♣rt② ♠♦♥tst♦ ♣rsr♥ t ♦♥tt s♦♥t♥ts ♥ ♦♥♠♥s♦♥ ♦s s♠s t♦ ♠♥t♦r② ♦r ♦t♥♥ r s♠s s ♠♥② t♦rs ♣r♦♣♦s♠♦ ♦♥♦ s♠s ♥ ♦rr t♦ ts ♣r♦♣rt② r♥ ♥ ❬r❪♣r♦♣♦ss t♦ s♦ t ♣rssr ♦t♦♥ qt♦♥ ♥st ♦ t ♠ss rt♦♥ ♦t♦♥ qt♦♥ t t ♥tr r ♥ r ❬ ❪ ♣r♦♣♦s t♦s♦ t ♠ss rt♦♥ qt♦♥ ♥ ♥♦♥♦♥srt ② ♥ ♦rr t♦ r♦r t♣rsrt♦♥ ♦ ♦♥st♥t ♦t②♣rssr stts ♣♣r♦ ♥ ♦♦rt♦rs ❬❪ ♥tr♦ t ♦st ♠t♦ t t ♥trt② ♣r♦♣♦s t♦ ♥tr♦ t♦ rt s ♥ ♦rr t♦ ♦♥strt s♠ tt♦♥② rqrs ♦♥ ♠♥♥ s♦r ♠t♦ s ♥ ♠♣r♦ ♥♠♥② ♦rs ❲ ♦♥♥trt r ♥ ♦♥ r♥t t ♦st t♦ ❬❲ ❪ t s ♥♦t ♣♦ss t♦ ♦♠♣r♥s sr②♦ ts t ♠♥② ♦tr tt♠♣ts ♥ ♣r♦♣♦s ♥♥ t t♦♠♦ t♦ ♠♦r ♥r ♦♥ ❬ ❲ ❪ s♥ r♥♥ ♣♣r♦t t ♥tr ❬❪ t

♦♠♠♦♥ tr ♦ t ♦♠♥t♦♥ ♣♣r♦s s tt t s♠s r♥r② ♥♦♥♦♥srt t s ♣♦ss t♦ ♦♥strt r② ①♦t s♠s tt r♦♥srt t t② r t♥ r② ♦♠♣t ♥ ♥ s ♦♥② ♦r ♠tst ss ❬❪ ♥tr qst♦♥ rss s tr ts s♠s

② ♦rs ♥ ♣rss ♥t ♦♠ ♦♥♦ s♠ ♦st ♠t♦ r♥♣r♦t♦♥ ♠♠ s♠ s ♦st♦♥s

P P P ❲P ❲

♦♥r ♦r ♥♦t t♦rs t r♥t s♦t♦♥ ♦ t ♥t t♦ ♠♦s ♥ ♥r② ♥♦♥♦♥srt s♠s ♦♥r t♦rs r♦♥ s♦t♦♥s t s ♣r② ♥♦♥♥r ♦r ❬❪ s st ♥♦t ②t ♥rst♦♦ s ♥♦♥♦♥srt ①❲♥r♦ t♦r② ♦s ♥♦t ①st ♦r r♥t ♦r ♦♥ ts s♣t s ❬❪ r t stt♦♥ s rtr st s t ♥♦♥♦♥srt♦♥♦ t s♠s s ♥r② ♦t t t ♦♥tt s♦♥t♥t② s ♥r②♥rt ❲♥ t s♦♥t♥♦s s s♦s ♥ ♦♥tts r s♣rt t s ts ♥♦t ♣r♦① t♦ ♦sr ♦♥r♥ t♦rs t ♦♦ s♦t♦♥ ♦r ♥ s ♦ ♦♠♣t ♥♦♥♥r ♥trt♦♥s ♥ t sr ♠① t s t t♦ ♥rst♥ ② t ♥♦♥♦♥srt ♣♣r♦ s t♦ ♦♥r♥ s♠s ♦r ♦♥♠♥s♦♥ ♣r♦♠s ♠①♥ ♦r ♥ r②s♠♣ stt♦♥s t t ♥t t♠ ♦ ♠♥♥ ♣r♦♠ ♦r ♥st♥ ♦r ♥ s♦ s s♥t ♦r ♠♦♥ ♥tr

r rst ♦t ♥ ts ♣♣r s t♦ ♣r♦ ♥ ♥♦♥♦♥srt s♠♦r s♦♥ t♦ ♦s r ♣♣r♦ s ♥ ♣tt♦♥ ♦ ♣r♦s ♦rs ♦♦t♥ ♦♥s ♥ ♦q ❬ ❪ ♦♥ r♥♣r♦t♦♥ s♠s s t♦ s ♣r♦t♦♥ st♣ s ♦♥ r♥♦♠ s♠♣♥ t♥qs r② s♠rt♦ t ♠♠ s♠ ♠t♦ ss ♠♠ s♠ ❬❪ ♠♣s ♥ ①t ♠♥♥ s♦r r s t r♥♦♠ s♠♣♥ s ♦♥② ♣r♦r♠ ♥ t♣r♦t♦♥ st♣ t s ♣♦ss t♦ r② ♦♥ ♣♣r♦①♠t ♠♥♥ s♦rs ♥ t r♥ st♣ ❲ s tt ♥ ♣rs♥ ♦ str♦♥ s♦s ♦r ♣♣r♦ s t♦ ♣t ♥ ♦rr t♦ ♦ ♦st♦♥s ♥ ♥♦♥♦♥r♥ s♠♣② ♣r♦♣♦s t♦♣r♦r♠ t r♥♦♠ s♠♣♥ strt② ♦♥② t t t♦ s ♥tr s ♦r ♦trs♠s ♦r r♥♦♠ s♠♣♥ ♣r♦t♦♥ s♠ s ♥♦t ♦♥srt t s rtt t ♠♠ s♠ s ♥♦t ♦♥srt t♦♦ t ♣♦sssss sttst② ♦♥srt♦♥ ♣r♦♣rts ❬❪ ❲ ♦♣ tt s ♣r♦♣rts st ♦ ♦r ♦r s♠♣♥♣r♦t♦♥ s♠

r s♦♥ ♦t s t♦ ♣r♦r♠ ♥♠r ♦♥r♥ st② ♦r srss ♥♦♥♦♥srt s♠s ♦r t♦ ♦s ♥ ♦♠♣r t♠ t♦ ♦r ♥s♠ ❲ ♦sr sr♣rs♥② tt t ♥♠r s♦t♦♥s s♠ t♦ ♦♥rt♦rs t ♦♦ s♦t♦♥s ♦r♥ t♦ ♦r ♣r♦s ♦♥srt♦♥s ts♦r s s♦t② ♥♦t ♦♦s ♥② ♦♠♣r ♦r ♥ s♠ t♦ t ♥ ♠♦r ♦♠♣① ♦♥rt♦♥ ❲ ♣rs♥t r tst s ♦♥sst♥ ♥♦♠♣t♥ t ♦st♦♥s ♦ s♣r s ♥ ♦♠♣rss q ❲♣rs♥t t rsts ♦t♥ t t ♥ t r♥♦♠ ♣r♦t♦♥ s♠

t♦ ♠♦

♥ ts ♣♣r ♥stt t ♥♠r rs♦t♦♥ ♦ t r s②st♠ ♦r ♦♠♣rss t♦ ♠①tr ♥st② ♦ t ♠①tr s ρ t ♦t② s u ♥t ♥tr♥ ♥r② s e ❲ ♥♦t ② E t t♦t ♥r② ♥ ② E = e+u2/2 ♣rssr s ♥♦t p ♦r s♠♣t② t t♦t ♦ss ♥rt② ♦♥②♦♥sr ♦♥♠♥s♦♥ ♦s ♥♥♦♥s ♣♥ ♦♥ t s♣t ♣♦st♦♥ x♥ ♦ t t♠ t P s②st♠ s ♠ ♦ ♠ss ♠♦♠♥t♠ ♥ ♥r②♦♥srt♦♥ s

∂tρ+ ∂x(ρu) = 0,

∂t(ρu) + ∂x(ρu2 + p) = 0,

∂t(ρE) + ∂x((ρE + p)u) = 0.

P P P ❲P ❲

♥ t s ♦ ♦♥ ♦ t ♣rssr ♦ ♥t♦♥ ♦ t ♥st②♥ t ♥tr♥ ♥r②

p = p(ρ, e).

s ♦♥sr t♦ ♦s ♦r ♣rssr s ♥t♦♥ ♦ t ♥st② ♥t ♥tr♥ ♥r② t s♦ ♦ s♣♣♠♥tr② ♥♥♦♥ ϕ t ♦♦r ♥t♦♥

p = p(ρ, e, ϕ).

♦♦r ♥t♦♥ s tr♥s♣♦rt t t ♦

∂tϕ+ u∂xϕ = 0.

♦♠♥♥ ts tr♥s♣♦rt qt♦♥ t t ♠ss ♦♥srt♦♥ s ♦♥srt ♦r♠ ♦ t ♦♦r ♥t♦♥ qt♦♥

∂t(ρϕ) + ∂x(ρϕu) = 0.

♥② ♥♥ t ♦♥srt rs t♦r

W = (ρ, ρu, ρE, ρϕ)T ,

♥ t ① t♦r

F (W ) = (ρu, ρu2 + p, (ρE + p)u, ρϕu)T ,

t s②st♠ ♥ rtt♥

∂tW + ∂xF (W ) = 0.

♦r ♣rt ♦♠♣tt♦♥s s t ♣rssr ♦ ♠①tr st♥s ❲ ♦♥sr s ♥ q sts②♥ st♥ s s

p = (γi − 1)ρe− γiπi,

t i = 1 ♦r t s ♥ i = 2 ♦r t q ♣r♠trs γi > 1 ♥ πi r♦t♥ r♦♠ ♣②s ♠sr♠♥ts ♠①tr ♣rssr s ♥ ②

p(ρ, e, ϕ) = (γ(ϕ)− 1)ρe− γ(ϕ)π(ϕ).

♠①tr ♣r♠trs r ♥ ②

1

γ(ϕ)− 1= ϕ

1

γ2 − 1+ (1− ϕ)

1

γ1 − 1,

γ(ϕ)π(ϕ)

γ(ϕ)− 1= ϕ

γ2π2γ2 − 1

+ (1− ϕ)γ1π1γ1 − 1

,

♥ s ② tt ϕ = 1 ♥ t ♣r q ♣s ♥ ϕ = 0 ♥ t ♣r s ♣ss s②st♠ s ♥ ♠t♠t ♣r♦♣rts t s ②♣r♦ ♥ t ♠♥♥♣r♦♠ s ♥q s♦t♦♥ ♥ t r t ❬❪

♥ t ♥♠r s t stt♦♥ s ♠♦r ♦♠♣t ♦r ♥st♥ t s ♥♦♥♦♥ tt st♥r ♦♥srt ♥t ♦♠ s♠s ♣♦♦r ♣rs♦♥♥ ♣♣ t♦ ts ♥ ♦ ♦ ♥ ♦rs ♥ s♦♠ ♦♥rt♦♥s ♦ qs♦s t ①♣t ♦♥♦ ♥♥♦t s s t s t♦ ♥t ♥sts

P P P ❲P ❲

r♥♣r♦t♦♥ ♣♣r♦

♦r t ♥t ♦♠ ♣♣r♦①♠t♦♥ ♦♥sr sq♥ ♦ t♠s tn n ∈ Ns tt t0 = 0 ♥ τn = tn+1 − tn > 0 ❲ s♦ ♦♥sr ♠s ♣♦♥ts xni+1/2 t

t♠ n Cni s t ♥tr ]xni−1/2, x

ni+1/2[ ❲ ♥♦t ② xni t ♥tr ♦

Cni

xni =xni−1/2 + xni+1/2

2.

♥t ♦ Cni s ♥♦t hni = xni+1/2 − x

ni−1/2 ♦r♥ t♦ t ♥♦tt♦♥s

t ♠s s ♠♦♥ t t s♦♠ t♠ st♣ ♦ t♦ t ♥t ♠s tn = 0 ❲ ♥♦t

xi = x0i , Ci = C0i , hi = h0i , etc.

❲ r ♦♦♥ ♦r ♥ ♣♣r♦①♠t♦♥ ♦ W ♥ t Cni

Wni ≃W (x, t), x ∈ Cn

i , t ∈]tn, tn+1[.

♦r t ♥♠r rs♦t♦♥ ♥ ♥ ①t ♦r ♣♣r♦①♠t ♠♥♥ s♦r ①t ♦r ♣♣r♦①♠t s♦t♦♥ ♦ t ♠♥♥ ♣r♦♠

∂tV + ∂xF (V ) = 0,

V (x, 0) =

WL x < 0,WR x > 0,

s ♥♦t

R(x

t,WL,WR) = V (x, t).

t♠ st♣ ♦ t r♥♣r♦t♦♥ s♠ s ♠ ♦ t♦ sts ♥ trst st ♣♣r♦①♠t t s♦t♦♥ t r♥ s♠

hn+1/2i W

n+1/2i − hni W

ni + τn

(Fni+1/2 − F

ni−1/2

)= 0.

r♥ ① s ♥ ②

Fni+1/2 = F (Wn

i+1/2)− uni+1/2W

ni+1/2,

Wni+1/2 = R(uni+1/2,W

ni ,W

ni+1),

r t ♦♥r② xni+1/2 ♠♦s t t ♦t② uni+1/2 ♦ t ♦♥tt s♦♥t

♥t② ♥ t rs♦t♦♥ ♦ t ♠♥♥ ♣r♦♠ t♥WL =Wni ♥WR =Wn

i+1

xn+1/2i+1/2 = xni+1/2 + τnu

ni+1/2.

♥ ♣rtr ts ♥s t ♥ s③ ♦ Cni

hn+1/2i = x

n+1/2i+1/2 − x

n+1/2i−1/2 = hni + τn(u

ni+1/2 − u

ni−1/2).

s ♦r♠ s ♠♣♦rt♥t s t ♥ ♥r③ t♦ r ♠♥s♦♥s t♣r♠ts t♦ ♦ t t ♦♠♣tt♦♥ ♦ t ♠♦ ♠s

tr t r♥ st t♦ ♦ t♦ t ♥t r ♠s s ♥ ♦♥ t sr ♠t♦s

P P P ❲P ❲

r♥ ♣r♦t♦♥ ♥ ts ♣♣r♦ r ♦♥ t rr t s♠♣ L2 ♣r♦t♦♥

Wn+1

i =τnhi

max(uni−1/2, 0)Wn+1/2i−1

−τnhi

min(uni+1/2, 0)Wn+1/2i+1

+

(1−

τnhi

max(uni−1/2, 0) +τnhi

min(uni+1/2, 0)

)W

n+1/2i .

♥ ♦ t♦ t ♥t r r

Cn+1

i = Ci, hn+1

i = hni .

t ♥ s♦ rtt♥

Wn+1

i =Wn+1/2i −

τnhi

(max(uni−1/2, 0)(W

n+1/2i −W

n+1/2i−1

)+

min(un+1/2i+1/2 , 0)(W

n+1/2i+1

−Wn+1/2i )

).

♥ ts ② t s r tt t ♣r♦t♦♥ st♣ s ♥ ♣♥ ♣♣r♦①♠t♦♥ ♦

∂tW + u∂xW = 0.

s ♠t♦ s ② ♦♥srt ♥ ts s ♣rs♦♥ ♦r ♠t ♣r♦♠s ❬❪ t s ♣♦ss t♦ ♠♣r♦ t ♣rs♦♥ ② t rr ♣♣r♦ t ♦♥ssts ♥ ♣r♦r♠♥ ♥♦♥♦♥srt ♣r♦t♦♥ ♦♥ t ♦♦r ♥t♦♥ ♥st ♦ ♣r♦t♥ ρϕ s ♥ ♣r♦t rt② ϕ s

ϕn+1

i = ϕn+1/2i −

τnhi

(max(uni−1/2, 0)(ϕ

n+1/2i − ϕ

n+1/2i−1

)+

min(un+1/2i+1/2 , 0)(ϕ

n+1/2i+1

− ϕn+1/2i )

).

s ♣♣r♦ rsts ♥ ♦② ♥♦♥♦♥srt s♠ t ♥s ♥♠r♠ss tr♥sr t♥ t t♦ ♣ss ♥ t s ♦ t st♥ s ♣rssr t ♥ ♣r♦ tt t rst♥ s♠ ♣rsrs ♦♥st♥t (u, p) stts

♠♠ ♣r♦t♦♥ ♥ ts ♣♣r♦ ♦♥strt sq♥ ♦ r♥♦♠ ♦r ♣s♦r♥♦♠ ♥♠rs ωn ∈ [0, 1]. ♦r♥ t♦ ts ♥♠r t

Wn+1

i =Wn+1/2i−1

ωn <τnhi

max(uni−1/2, 0),

Wn+1

i =Wn+1/2i+1

ωn > 1 +τnhi

min(uni+1/2, 0),

Wni =W

n+1/2i

τnhi

max(uni−1/2, 0) ≤ ωn ≤ 1 +τnhi

min(uni+1/2, 0).

♥ ♦ t♦ t ♥t r r

hn+1

i = hni .

s ♠t♦ s ♦♥② sttst② ♦♥srt ❬❪ t ♣rsrs ①t② ♦♥st♥t♦t②♣rssr stts ♦♥tts r s♦ ♥ ♦♥ ♣♦♥t r stt t s♦t♦♥ ♠② ♥♦s② ♥ ♣rtr ♦r str♦♥ s♦s t ♦ ♠t♦♦s ♥♦t ♦♥r t♦rs t ♦rrt ♥tr♦♣② s♦t♦♥

♦♦ ♦ ♦r t ♣s♦r♥♦♠ sq♥ ωn s t (k1, k2) ♥ r ♦r♣tsq♥ ♦♠♣t ② t ♦♦♥ ♦rt♠

P P P ❲P ❲

r t ♦ t ♦♠♣tt♦♥ ♦ t r ♥ ♦st stts r♦♠ t ♥tr stts uI pI ♥ ρIL ρIR tr♠♥② s♦♥ t♦♣s ♠♥♥ ♣r♦♠ ♦r t stts uL ♥ uR

♦t ♦r♣t♥t ♥♥t ♥t ④

♦t ♦r♣t

♦t s

♥④

s

♦r♣t♥s

♥

⑥

rtr♥ ♦r♣t

⑥

♥ ts ♦rt♠ k1 ♥ k2 r t♦ rt② ♣r♠ ♥♠rs ♥ k1 > k2 > 0 ♦r♠♦r ts rr t♦ ❬♦r❪ ♥ ♣rt ♦♥sr t (5, 3) ♥ r ♦r♣tsq♥

① ♣r♦t♦♥ ♥ ♦rr t♦ ♠♣r♦ t ♦♥r♥ ♦ t ♠♠ ♣♣r♦ t s ♣♦ss t♦ ♦♦ t ♦♦♥ ♠① ♣r♦t♦♥ st♣ Ci ♥ts t♦ ♥♦rs r ♥ t s♠

(ϕni−1 −

1

2)(ϕn

i −1

2) > 0 ♥ (ϕn

i −1

2)(ϕn

i+1 −1

2) > 0,

t♥ ♦♦ t ♣r♦t♦♥ ♥ ② ♥ t ♦tr ss ♦♦ t♠♠ ♣r♦t♦♥ s ♣♣r♦ ♦s ttr ♣rs♦♥ t t ♥trs t s rs♦ ♥ ♦♥② ♦♥ ♣♦♥t

♠♦ ♦st ♣♣r♦

r ♦st ♠t♦ ♦♣ ② ❲♥ ♥ ♦♦ ♥ ❬❲❪ s ♥ ♣tt♦♥ ♦ t ♦r♥ ♦st ♠t♦ ♦

♥ ts ♠t♦ t ♥tr t♥ t q ♥ t s s ♦t ② ♥t♦♥ ψ ♥ t q ψ > 0 ♥ ♥ t s ψ < 0 ♥ ts

P P P ❲P ❲

t ♥tr ♦rrs♣♦♥s t♦ t st ψ = 0 s ♥ t ♣r♦s ♠t♦ tst ♥t♦♥ ψ s tr♥s♣♦rt ♥ t ♦

∂tψ + u∂xψ = 0.

♥ st r♦♠ t ♣rssr ♦ t q t♦ t ♣rssr ♦ ts ♦r♥ t♦ t s♥ ♦ ψ t s r tt t r♥s t♥ t ♦♦r♥t♦♥ ♠♦ ♥ t st ♠♦ r ♦♥② ♦r♠ ♦r t ♥♠r♠♣♠♥tt♦♥s r rtr r♥t

st ♥t♦♥ ψ s ♣♣r♦①♠t ♥ Ci t t♠ tn ② ψni s♦t♦♥

s ♣♣r♦①♠t ② ♦♥♦ s♠

Wn+1

i =Wni −

τnhi

(Fn,−i+1/2 − F

n,+i−1/2

),

t ♣♦ss ♥♦♥♦♥srt ① Fn,−i+1/2 6= Fn,+

i+1/2 t t♦ s Ci ♥ Ci+1

r t t s♠ ♣s s tr

ψni · ψ

ni+1 > 0.

♥ t t ss ♦♥srt ♦♥♦ ①

Fn,−i+1/2 = Fn,+

i+1/2 = Fni+1/2 = F (R(0,Wn

i ,Wni+1)).

t ♣s ♦♥r② s ②♥ t♥ i ♥ i+1 r i ♦rrs♣♦♥st♦ A ♥ i+ 1 t♦ B t♥ t ♠♥s tt

ψni · ψ

ni+1 < 0.

t♥ t t ♥ rt stt r t♥ r♦♠ s i − 1 ♥ i + 2 rs♣t② t♦♥sr ss t♦ stts ♦ t ♣r ♣ss

WL =Wni−1, WR =Wn

i+2.

❲ s♦ t ♠♥♥ ♣r♦♠ t♥WL ♥WR t uI t ♦♥tt ♦t②♥ ts ①t s♦t♦♥ ❲ ♥ ♥ ♥tr stts t♦ t t ♥ t♦ t rt♦ t ♦♥tt ②

WIL = limξ→u−

I

R(ξ,WL,WR), WIR = limξ→u+

I

R(ξ,WL,WR).

❲ ts ss t♦ ♥tr stts ♦r ♥sts ρIL ρIR ♣rssr pI ♥♦t② uI t ♥ rt t♦ t ♣s ♦♥r② ♦r A t stt (ρIL, uI , pI)r♣s t stts ♦ t ♥ ♥s t ♦st stts ①s r ts

F−

i+1/2 = F (R(0,WIL,WIL)), F+

i+1/2 = F (R(0,WIR,WIR)).

♦t♦♥ qt♦♥ s s♦ st② ♠♦ ②

Wn+1

i =WIL −τnhi

(Fn,−i+1/2 − F

n,+i−1/2

),

♥

Wn+1

i+1=WIR −

τnhi+1

(Fn,−i+3/2 − F

n,+i+1/2

),

s ♣r♦r s st ♥ s ♦♥sq♥ ♦♥② s♥♣s ♠♥♥♣r♦♠s r s♦ ♦r ♥tr ♦ A t♦ ♣r♦ t ♥♠r ①st t ♦st s s ♦♥r② ♦♥t♦♥s t t ♣s ♦♥r② ♥ t s♠♣r♦r s s ♦r B

s ♥r t ♣s ♦♥r② t♦ ①s Fn,±

i+ 12

♦♥ ♦r r ♥

② r♦♠ t ♣s ♦♥r② r ♦♥② ♦♥ ♥♠r ① s ♦♠♣t t

P P P ❲P ❲

♥tr t s♣t ♦rr s ♠♣r♦ ② s♥ s♦♥♦rr r♦♥strt♦♥♦ t ♣r♠t rs ρ u p s♦t♦♥ s ♥ t♦ t ♥①t t♠ st♣② t ♥t ♦♠ s♠

♥ t ♦tr ♥ t st ♥t♦♥ s s♦ t♦ ♥ s s ♦♥rst ② s♦♥ ♥♠r② t st♥r ♣♥ ♥♦♥♦♥srt ♥t♦♠ s♠

ψn+1,−i = ψn

i −τnhi

(max(uni−1/2, 0)(ψ

ni − ψ

ni−1) + min(uni+1/2, 0)(ψ

ni+1 − ψ

ni ))

Pr♦② t st ♥t♦♥ ♣♣r♦①♠t♦♥ s r♥t③ ♥ s ②tt t r♠♥s s♥ st♥ t♦ t ♥tr s s ♦r♠② ♦t♥ tr♦t ♥♠r rs♦t♦♥ ♦ ♥ ♠t♦♥♦ qt♦♥

∂τ ψ(x, τ) + a(ψ)∂xψ = S(ψ),

a(ψ) = S(ψ)∂xψ∣∣∣∂xψ

∣∣∣,

S(ψ) =

−1 ψ < 0,

0 ψ = 0,

1 ψ > 0,

ψ(x, τ = 0) = ψn+1,−i , x ∈ Ci.

st ♥t♦♥ s r♣ ② t r♥t③ st ♥t♦♥ t

ψn+1

i = ψ(x, τ =∞), x ∈ Ci.

s ♣r♦r s sr ♥ ♠♦r ts ♥ ❬❪♥② r♥ t ♣t ♦ t st ♥t♦♥ ♠② st r♦♠ ♦♥

t♦ t ♦tr s stt♦♥ ♦rrs♣♦♥s t♦ ♥ ♦ t s♥ t♥ t♠st♣n ♥ t♠ n + 1 ♥ ψn

i · ψn+1

i < 0 ♥ ts s t s ♥ssr② t♦

s♦ ♣t Wn+1

i ♦♥ t ♦rrs♣♦♥♥ rs r rts♥ t qt♦♥ ♦ stt ♦ t ♥ ♥ r② ♦s t♦ t ♣s♦♥r② t ♦ts ♥ t ♣rssr r ♦♥st♥t ♦r ♦t s t t♣s ♦♥r② r ♣rsr s ♠♦t♦♥ s sst ② rr♦♥❬❪♥ t♦♥ t♦ ts ♣♣r♦ ♣r♦♣♦s t ♠♦t♦♥ ♦ t ♥st② s ♥♦①t ♦r t ♥st② s ♥♦♥ t ♥st② s r♣ ② t ♥st② ♦ t♦rrs♣♦♥♥ ♦st ♦r ♣rs② ψn

i · ψn+1

i < 0 ♥ ψni · ψ

ni+1 < 0

t♥ ♦r ♦♠♣t♥ t ♥①t t♠st♣ ssttt t ♥st② ②

ρn+1

i ← ρIR

♥ t ♥r② en+1

i s s♦ ♠♦ ♥ s ② tt

pn+1

i = p(ρn+1

i , en+1

i , ψn+1

i )

s ♥♦t ♥ s ♦♥strt♦♥ ♠♣s tt t ♦ rst♥ s♠ ♣rsr ♦♥st♥t (u, p) stts ♥ t ♦tr ♥ t s s♦ r tt t s♠s ♥♦t ♦♥srt ♦r ♥st♥ t st ♣t ♠♣s ♠ss ♥ ♥ ♥r②tr♥sr t♥ t t♦ s

P P P ❲P ❲

r ♦♥r♥ st② ♠♠ ♣r♦t♦♥ rss r♥ ♣r♦t♦♥ ♠ t♦♥

♠r rsts

♠ t♦♥ rst tst ♦♥ssts ♥ t♦ s♦ t st♥ s ♣r♠tr r

γ2 = 2, π2 = 1,

γ1 = 1.4, π1 = 0.

❲ t ♦r t t ♥ rt ♥t t

(ρL, uL, pL, ϕL) = (2, 1/2, 2, 1),

(ρR, uR, pR, ϕR) = (1, 1/2, 1, 0).

♦r t r♥♣r♦t♦♥ ♣♣r♦ t ♥♦♥♦♥srt ♣r♦t♦♥ ♥ t♠♠ ♣r♦t♦♥ r ♦♠♣r ❲ ♦sr ♥♠r ♦♥r♥ ♥ t L1

♥♦r♠ ♦r t t♦ ♠t♦s ♥ tt t ♠♠ ♣r♦t♦♥ s ♠♦r ♣rs t♥ tr♥ ♣r♦t♦♥ r ♦♥r♥ rt ♦r t t♦ ♠t♦s s♣♣r♦①♠t② 0.6

♠ s♦♥tr ♥trt♦♥ ♥ ♥tr t♥ t♦ ss ♦t t♠ t = 0 t ♣♦st♦♥ x = 1 t♦ s r ♠♦♥ t♦ t tt t ♦t② v = −1 s ♦♥ t t t s ♦♥ trt s♦ s rr♥ r♦♠ t t t ♦t② σ = 4 ♥t ♣♦st♦♥ ♦

P P P ❲P ❲

r ♠♠ ♣♣r♦ ♥st② ♣♦t ❱ ①♣♦s♦♥ t♦t♥ t ♣r♦♣t♦♥

t ♦♥tt ♥ t s♦ r ♦s♥ ♥ s ② tt t② ♠t t♦tr tt sss x = 0 t t♠ t = 1 ♣r♠trs r t ♦♦♥

γ1 = 1.4 π1 = 0,

γ2 = 2 π2 = 7.

♥t t r x < −4

(ρL, uL, pL, ϕL) = (3.4884, 1.1333, 23.333, 1),

x > 1

(ρR, uR, pR, ϕR) = (1,−1, 2, 0),

♥ −4 ≤ x ≤ 1

(ρM , uM , pM , ϕM ) = (2,−1, 2, 1).

tr tt t s♦ ♥ t ♦♥tt s ♠t t t♠ t = 1 t s♦t♦♥ ss♠♣② ♥ ② t rs♦t♦♥ ♦ t♦ ♠♥♥ ♣r♦♠ t♥ stts (L)♥ (R) s♦t♦♥s s st ♥ r ♥♠r t r r♥

r♥ ♣s ♣r♦t♦♥ s♠s ♥ ts s ♦sr tt t ♠♠♣♣r♦ ♦s ♥♦t ♦♥r s ♦r ♣♥s ♦♥ t str♥t ♦ t s♦ t②♣ ♣♦t s ♥ ♦♥ r r ♦♠♣r t ①t ♥ t♣♣r♦①♠t ♥sts t t♠ t = 1.5

♥ ts s ts ♦♠♣r t r♥ ♣r♦t♦♥ ♣♣r♦ t t ♠①♣r♦t♦♥ ♣♣r♦ ❲ ♦t♥ t rsts ♦ r ♠① ♣r♦t♦♥s ttr ♣rs♦♥ t♥ t r♥ ♣r♦t♦♥

❲ s♦ ♣r♦ ♦♥ r ♦♠♣rs♦♥ ♦ t ♠① ♥ r♥ ♣r♦t♦♥s♠s ♦r t ♥sts ♦r ♠s ♦ s ♦ t ♥tr [−5; 2]

P P P ❲P ❲

r ♠ s♦♥tr ♥trt♦♥ ♦♥r♥st② ① ♣r♦t♦♥ ♥ r♥ ♣r♦t♦♥

♥♠r s ① t♦ 0.7 t s ♥trst♥ t♦ ♦sr tt t ♥tr ♣♦st♦♥ sr② rs♦ ♥ ♦♥② ♦♥ ♠s ♣♦♥t ② t ♠① ♣r♦t♦♥ s♠ ♥tt ts ♦♦ rs♦t♦♥ ♦ t ♦♥tt s♦ ♠♣s ♥ ♠♣r♦♠♥t ♦ t♣rs♦♥ ♥ t t rrt♦♥

♦ ♦st ♣♣r♦ ♥ ♦rr t♦ ♦♠♣r t ♥♦♥♦♥srt♠t♦s ♦ rr ♥ t sr ♥♠r s♦t♦♥s r ♦♠♣r t t ①t s♦t♦♥ ♥ ♦♥r♥ st② s ♣r♦r♠ ♦rssrt③t♦♥ ♦♥ssts ♦ s ♦♥ t ♠tss tr♥s♦r♠t♦♥ s♣♣ ❬❪ ♦♥r♥ st② s ♣r♦r♠ ♦r rs ♥ r♦♠ t♦ r♥♠♥t s L t ♥♦r♠ r ♦♥ t ♥st ♦♥ssts ♦ 2L ∗100s trs♦ ♥ t r ♣tt♦♥ s ♦s♥ s ε = 10−5 s s♠ s ♦s♥ ♥ s ② tt t r♦♥s ♦♥t♥♥ t ♥st r s rr ♥♦ t♦ ♦ t♦♥ rr♦r r♦♠ rr s rr♦rs ♦t♥ tt ♠ts r ♣tt♦♥ r ts ♦♠♣r t t♦s ♦t♥ t ♥♦r♠ r sts r ♣r♦r♠ t ♥♠r ♦ 0.9 ♠♠r②Ω❬❪♠t ∈❬❪s N0 = 100 5 ≤ L ≤ 13 ǫL

♥ rs ♥ r s♦♥ t ♦♠♣rs♦♥ ♦ t ♥st② ♦r t t♦ ♣♣r♦s t t ①t s♦t♦♥ t s ♥ t t = 1.5 ♠s rst r♦rrs♣♦♥s t♦ r t s♥ r♥♠♥t s ♥ t s♦♥ ♦♥ t♦ r② ♥r t t r♥♠♥t s ♦ rst s ♣tr ♥ t ♠ ♥

P P P ❲P ❲

r ♥st② ♦♠♣rs♦♥ ♦ t ♠① ♥ r♥♣r♦t♦♥ s♠s

r ♥t③t♦♥ ♦ t rr s♦♦♥tt ♥trt♦♥

UW UWS UW∗ UA∗ UA

ρ ❬♠3❪ v ❬♠s❪ p ❬P❪ rr s♦♦♥tt ♥trt♦♥

③♦♦♠ ♦ t s♦ ♣♦st♦♥ ♦♥ t ♦tt♦♠ t ③♦♦♠ ♦ t ♣t t♥t rrt♦♥ ♥ t ♦♥tt ♦♥ t t♦♣ t ♥ ③♦♦♠ r② ♦s t♦ t♦♥tt ♣♦st♦♥ ♦♥ t rt ♥ t st ♦♥ ♥trs r ♠r ② ♠♦♥s♦r t ♥♠r rsts

rr ♣♣r♦ ♥rts ♦st♦♥s t t ♦tt♦♠ ♦ t rrt♦♥ t♦ t ♥trt♦♥ t♥ t s♦ ♥ t ♦♥tt s s ♥♦tt s ♦r t tt ♦♥s qt t t ①t s♦t♦♥ ❲t ♠♦r r r♥♠♥t s s♦♥ ♥ ♣tr t ♠♣t ♦ t ♦st♦♥sr② r

t t ♦♥tt tr s s♠r♥ ♦ t ♥st② ♦r t rr ♣♣r♦s s s② t♦ s ♥ t ③♦♦♠ ♦♥ t rt ♥ t ♥st② ♦r ts ♣♣r♦rss s♦② ♥st ♦ ♣rs♥t♥ ♠♣ t t ♠t♦ t♦ t

P P P ❲P ❲

r sts ♦ t s♦♦♥tt ♥trt♦♥ ♦r r♥♠♥t s t t ♠s

♦♥strt♦♥ ♦ t ♦st ♠t♦ ♦t♥ t sr ♠♣ t t ♦♥ttt ts ♠♣ s tt t st ♦♠♣r t t ①t s♦t♦♥ s t ♥trs r r♣rs♥t ② ♠♦♥s ♦♥ t ♣trs ♦r r♥♠♥t s ♥ s st ♥ t ♣♦st♦♥ ♦ t ♦♥tt ♦ s ♥ ♥ t ♦♠♣tt♦♥ tt s ♦ r♥♠♥t t st s ♦t s s s s♠♠r③ ♥

♦♥r♥♥ t ♣♦st♦♥ ♦ t s♦ s ♦♥② s ♥ t ③♦♦♠ s♦ t ♠♣ ♦ ♥st② r♠r tt ts ♣♦st♦♥ s ♣rt t t ♠t♦ ♦r ♦t ♦♠♣tt♦♥s t s ♥♦t r② t s ♦r t rr ♣♣r♦ ❯♥r r r♥♠♥t t st rs s ♠ s t s♠r♥r♦♥ t t ♦♥tt

♦♥r♥♥ t rr ♣♣r♦ t ♦st♦♥s ♥ t rr♦r ♥ t♣♦st♦♥ ♦ t s♦ s ♦♥② t♦ t ♣r♦s s♦♦♥tt ♥trt♦♥ ❲♥♦♥② t ♠♥♥ ♣r♦♠ s ♦♠♣t ♥ t ♦♠♣tt♦♥ strts t t = 0.5♠s ♦t♥ t rsts ♦ r ♥ ts s♠♣ stt♦♥ t rsts r ♥♦♦ r♠♥t r②r t t ①t s♦t♦♥ ♦♠♣rs♦♥ s♦♦♥tt♥trt♦♥ ♥ rt rs♦t♦♥ ♦ t ♠♥♥ ♣r♦♠ s s♦♥ ♥ r

L1 rr♦r ♦ t ♥st② r ♥ ♥ ♦rr ♦ ♦♥r♥ ♦rt rr ♣♣r♦ s ♣♣r♦①♠t② 0.5 ♦♥r♥♥ t t rr♦rs♠s t♦ t♥ t♦ t s♠ ♦rr ♥r r r♥♠♥t

rtr tst s ♦r t ♣r♦♣♦s ♥ t♦♥ tst s♦♥sst♥ ♥ s♦ ♦ ♥♠r 0.67 t t ♣♦st♦♥ x = −3 ♠ r♥♥♥♥ t q tt ♥trts t t r t t = 0.5♠s t t ♣♦st♦♥ x = 0 ♠ tr ♦ t s♦ ♥ t r r ♠♦♥ t♦rs t s♦ t t ♦t②♦ 100 m/s ♦♠♣tt♦♥ ♦♠♥ s [−4; 2]♠ tst s st ♥ r

P P P ❲P ❲

r sts ♦ t s♦♦♥tt ♦r r♥♠♥t st t ♠s

r sts ♦r r♥♠♥t s t t ♠s

♥ t r♥t stts r ♥ ♥ ♠tr ♣r♠trs ♦r ts r st ♥

P P P ❲P ❲

rr s L1 rr♦r rr L1 rr♦r rr5 7 8 9 10 11 12 13

❲trr ♦ L1 rr♦r

rr s hL ❬♠❪ rr♦r rr rr♦r rr

❲trr rr♦r ♥ t ♥tr ♣♦st♦♥ r hL s tr s③ ♦r L r♥♠♥t s

UW UWS UW∗ UA∗ UA

ρ ❬♠3❪ v ❬♠s❪ p ❬P❪

❲trr s♦♦♥tt ♥trt♦♥

γ ❬❪ π ❬P❪❲tr r

tr ♣r♠trs ♦r tr ♥ r

s ♦st♦♥s

♥ ts st♦♥ ♣♣② t r♥♦♠ ♣r♦t♦♥ s♠ t♦ ♦st♦♥s tst s sr ♥❬ ❪ s♣t t qs ♦♥♠♥s♦♥ r♠♦r t tst s ♠♣s r② ♦♥ ♦♠♣tt♦♥s ♥ r② ♥♠ss ❲ ♦♠♣r t t rsts ♦t♥ t t ♦♥ rtrr② r♥rs

P P P ❲P ❲

r ♥t③t♦♥ ♦ t trr s♦♦♥tt ♥trt♦♥

♦♥s♦♥

♥ ts ♣♣r ♣r♦♣♦s ♥ s♠ ♦r ♦♠♣t♥ t♦ ♦s ♣rssr ♦st♦♥s t t ♥tr r ♦ t♥s t♦ r♥ ♥♣r♦t♦♥ ♣♣r♦ ♥ t r♥ st♣ t ♦♥tt s r ♣rt② rs♦♥ t ♥tr s ♥♦t s♠r ♥ t ♣r♦t♦♥ st♣ ♠♣♦② r♥♦♠s♠♣♥ strt② rst♥ s♠ ♣rsrs t ♦♥st♥t ♦t②♣rssrstts ♥ t ♥tr s s♦ t♥ ♦♥ r ♣♦♥t

♦ ♣♣r♦ ♣r♦r♠s ♦r s♦s t ♥ ♣rs♥ ♦ str♦♥s♦s t ♣♣rs t♦ ♦st♥ r♦r t♦ ♣t t ♣r♦t♦♥ st♣♥ ♦♥② ♣♣② t t t t♦ ♥tr s ♦t t♥s t♦ t ♠♣s♦ t ♦♦r ♥t♦♥ ❲ ♣r♦♣♦s t♥ ♥♠r rsts tt ♠♦♥strt t♦♦ ♦♥r♥ ♦ t s♠ s♣t tt t s ♥♦t ♦♥srt ❲ sr♣rs♥②♦sr ts ♦♥r♥ ♣r♦♣rt② ♦r ♦tr ♥♦♥♦♥srt s♠s ♦r t♦♦s

♥② ♣♣② ♦r s♠ t♦ ♠♦r ♥♥ ♣r♦♠ ♦♥ssts ♥t s♠t♦♥ ♦ t ♦st♦♥s ♦ s ♥ ♦♠♣rss q r s♠♣s♠ s ♦♦ rsts ♥ ts ♣rs♦♥ s ss t♥ t ♠♦r s♦♣stt ♦♣ t rtrr② ♠s r♥♠♥t

r ♣r♦s♣ts r ♥ sr rt♦♥s

• rst ♦ t♦ ♠♣r♦ t ♣rs♦♥ ♦ t r♥♦♠ ♣r♦t♦♥ s♠ rst ♦♦s ② t♦ ♦ t s t♦ ♦♣ t t s♦♥ ♦rr ❯①t♥s♦♥ s ①t♥s♦♥ s t♦ tt t t ♥tr ♥ ♦rrt♦ ♦ ♦st♦♥s ♦r t s♣r ♦♠♣tt♦♥s ♥♦tr ②t♦ ♠♣r♦ t ♣rs♦♥ s t♦ ♠♦② t s♠ ♥ ♦rr tt t ♦♠s♥ s ♥ ♦♥ ② ♣t♥ t ♠t♦ sr ♥❬❪

• ♥♥ ①t♥s♦♥ ♦ ♦♥sst ♥ ①t♥♥ t r♥♦♠ ♣r♦t♦♥s♠ t♦ t♦ ♦r tr♠♥s♦♥ ♦♠♣tt♦♥s s ♦ tr ♦r①♠♣ ② s♠♣ rt♦♥ s♣tt♥ ♦rt♠ s t ♦t♦ ♦rt♦♠♥ ♦r

r♥s

❬❪ r ♥rst♦♥ ♦ t r♦ s♠ ♦r t ♦♠♣tt♦♥ ♦ ♠①tr ♦ ♣rtss r ér♦s♣t

❬❪ é♠ r ♠r r♥ ♦♠♠♥t ♦♥ t ♦♠♣tt♦♥ ♦ ♥♦♥♦♥srt ♣r♦ts ♦r♥ ♦ ♦♠♣tt♦♥ P②ss

❬❪ rr♦♥ P ② ♥ ♦② Prt ♦♠♣tt♦♥ ♦ ①s②♠♠tr ♠t♦s ♥tr♥t♦♥ ♦r♥ ♦ ♥t ❱♦♠s

❬❪ ♦♥s ♦q ♣tr♥ ♥♥t② sr♣ srt s♦ ♣r♦s t t ♦♥♦ s♠ ②♣r♦ ♣r♦♠s t♦r② ♥♠rs ♣♣t♦♥s ♣r♥rr♥

P P P ❲P ❲

❬❪ ♦♥s rst♦♣ ♦t♥ P♦ r♥s♣♦rtqr♠ s♠s ♦r ♦♠♣t♥ ♦♥tt s♦♥t♥ts ♥ tr ♦ ♠♦♥ ♦♠♠♥ t ♥♦

❬❪ ♦♥ P s♠ rq rr♠♥ rr② sr t♥② ♥♦♥♦st♦r②r♥ ♣♣r♦ t♦ ♥trs ♥ ♠t♠tr ♦s t ♦st ♠t♦ ♦♠♣t P②s ♥♦

❬❪ ♠♠ ♠s ♦t♦♥s ♥ t r ♦r ♥♦♥♥r ②♣r♦ s②st♠s ♦ qt♦♥s♦♠♠ Pr ♣♣ t

❬❪ P ② rr ts ♥ ♣♣r♦①♠t ♠♥♥ s♦r ♦r rr♦ssst♦♥ ♦♠♣rss ♦s Pr♦ ♥

❬❪ ② P♣♣ ts éè♥ ür r ♥ r♥ ♣♣r♦ ♦r t♦♠♣t♥ ♦ ♦st♦♥s ♥t ♦♠s ♦r ♦♠♣① ♣♣t♦♥s ❱ ♦♥♦♥

❬❪ ♦ ♦♠s ❨ ♦ P♣♣ ❲② ♥♦♥♦♥srt s♠s ♦♥r t♦ r♦♥s♦t♦♥s rr♦r ♥②ss t ♦♠♣ ♥♦

❬r❪ r♥ ♠r t♦♠♣♦♥♥t ♦ t♦♥s ② ♦♥sst♥t ♣r♠t ♦rt♠ ♦♠♣t P②s ♥♦

❬❪ ♦ ♦tèr ♥ ♥ts ♥♠r s♠ ♦r t s♠t♦♥ ♦ ♥trs t♥ ♦♠♣rss s ② ♠♥s ♦ qt♦♥ ♠♦ ♦♠♣t P②s ♥♦

❬❪ ts t té♦rq t ♥♠érq s é♦♠♥ts tr♥st♦♥ ♣sP tss ❯♥rsté trs♦r tt♣trs♦rtsrtr

❬❪ ür ♠♥♥ rö♥♥r r③ P ② ♦♠♣rs♦♥ ♥ t♦♥ ♦ ♦♠♣rss ♦ s♠t♦♥s ♦ sr♥ tt♦♥ s ♦♠♣trs s

❬❪ r r r é♠ s♠♣ ♠t♦ ♦r ♦♠♣rss ♠t ♦s ♦♠♣t ♥♦

❬❪ r r r é♠ ♠t♣s ♦♥♦ ♠t♦ ♦r ♦♠♣rss ♠t ♥ ♠t♣s ♦s ♦♠♣t P②s ♥♦

❬♦r❪ ♦r♦ tr♦ ♠♥♥ s♦rs ♥ ♥♠r ♠t♦s ♦r ②♥♠s ♣rt ♥tr♦t♦♥ ♦♥ t♦♥ ♣r♥r❱r r♥

❬❲❪ ❲rs r♦♥ ♦r♥ rr② ② ♦♥srt ♠♦ ♦r ♦♠♣rss t♦♦ t ♦♥r♥ ♦♥ ♠r t♦s ♦r ②♥♠s Prt ♥tr♥t ♠r t♦s s ♥♦

❬❲❪ ❲♥ ❲ ♦♦ r ♦st ♠t♦ ♦r t s♠t♦♥ ♦♠t♠♠ ♦♠♣rss ♦ ♦♠♣t ♥♦

❲ ♥ ♥ ❯♥rsté trs♦r

♠ rss ②♠t♥strr