La méthode IIM pour une membrane immergée dans un fluide incompressible

u∗

Un k

DT

Δμun+1 2

u∗

Ω p u v p u v Ω (i, j) ∂/∂x X(s) s Γ Ω f : R → R xi [x, x + h] M + 1

x x+h s₁, s₂, ..., sM X(s1) x X(s2) Iv u Ipu Iv p N = 24 N = 12 12 ×12 u X_kn u Γn _u u(X(s, nΔt), nΔt) p u∗ _u∗ _{≈ u}n+1 ∇ · u∗ (i, j) T T Γn _Γn+1

x = (x1, x2, (n + 1/2)Δt) Ω_ε 2ε Γ (η, ξ) X(s) n(s) τ (s) n Γn+1 2

u p μ

μ p(x, t) u(x, t) Du Dt + ∇p = μΔu + G + F , ∇ · u = 0, G F F F

μ Ω = [0, 1] × [0, 1] u p Du Dt + ∇p = μΔu + G, ∇ · u = 0. u p p n + 1/2 u n + 1 u∗ Ω u∗ u∗ i u∗s

u∗ un+1 u∗ _un+1 _{= u}∗ s u∗ i Δtφn+1 Δt n n + 1 u∗ _{= u}n+1_{+ Δt∇φ}n+1_. φn+1 3/2 n + 1/2 ∂un+12 ∂t + u n+1 2 · ∇un+12 + ∇pn+12 = μΔun+12 + Gn+12_, ∂un+12/∂t Δun+1 2 un+1_{− u}n Δt = ∂un+12 ∂t + O(Δt 2_). (Δu)n+1+ (Δu)n 2 = Δun+ 1 2 + O(Δt2). un+1 2 · ∇un+12 3 2(u · ∇u) n₋ 1 2(u · ∇u)n−1= (u · ∇u)n+ 1 2 + O(Δt2).

un+1_{− u}n Δt + (u · ∇u)n+ 1 2 + ∇pn+12 = μ 2  Δun+1_{+ Δu}n_{+ G}n+1 2. u∗ un+1 _u∗ un+1_{= u}∗_{− Δt∇φ}n+1_, φn+1 ∇ · un+1= 0 ∇ · un+1_{= ∇ · u}∗_{− ΔtΔφ}n+1 _{= 0,} ∇ · u∗ _{= ΔtΔφ}n+1_. un+1 _(u∗_{− Δt∇φ}n+1₎ u∗ _{− Δt∇φ}n+1_{− u}n Δt + (u · ∇u)n+ 1 2 + ∇pn+12 = μ 2  Δ_u∗_{− Δt∇φ}n+1_{+ Δu}n_{+ G}n+1 2, u∗ _{− u}n Δt + (u · ∇u)n+ 1 2 + ∇  pn+12 − φn+1+μΔt 2 Δφn+1  = μ 2 (Δu∗+ Δun) + Gn+ 1 2. pn+12 u∗ _pn+1 2 φn+1 qn+12 pn+12 pn+12 = qn+12 + φn+1− μΔt 2 Δφn+1

φn+1 pn+12 u∗_{− u}n Δt + (u · ∇u)n+ 1 2 + ∇qn+12 = μ 2(Δu∗+ Δun) + Gn+ 1 2 u∗ qn+12 = pn−12_. Ω (i, j) (ih, jh) h Ω f ∈ C2(R) x f (x + h) = f (x) + hf(x) + h 2 2 f(x) + O(h3)

f (x− h) = f(x) − hf(x) + h 2 2 f(x) + O(h3). f(x) = f (x + h)− f(x − h) 2h + O(h2), f(x) = f (x− h) − 2f(x) + f(x + h) h2 + O(h). φn+1 ∇ · un+1 _{= 0,} un+1_{= u}∗_{− Δt∇φ}n+1_. (i, j) p,u, u∗ φn+1 un+1 vn+1 un+1 un+1_i,j = u∗_i,j− Δtφ n+1 i+1,j − φn+1i−1,j 2h vn+1_i,j = v_i,j∗ − Δtφ n+1 i,j+1− φn+1i,j−1 2h , un_i,j u(ih, jh, nΔt)

un+1_i+1,j − un+1_i−1,j+ v_i,j+1n+1 − v_i,j−1n+1

2h = 0.

un+1

u∗_i+1,j − u∗_i−1,j+ v∗_i,j+1− v∗_i,j−1

2h

= Δtφn+1i+2,j + φn+1i−2,j+ φn+1i,j+2+ φn+1i,j−2− 4φn+1i,j

(i + 2, j) Δφn+1 i,j (i+1, j) φn+1 u∗ i+1 2,j− u ∗ i−1 2,j+ v ∗ i,j+1 2 − v ∗ i,j−1 2 h

= Δtφn+1i+1,j + φn+1i−1,j+ φn+1i,j+1+ φn+1i,j−1− 4φn+1i,j

h2 . i j u∗ un+1 (i +1₂, j) v∗ vn+1 (i, j + 1 2) p φn+1 (i, j) un+1 i+1 2,j = u ∗ i+1 2,j− Δt φn+1_i+1,j − φn+1_i,j h vn+1 i,j+1 2 = v ∗ i,j+1 2 − Δt φn+1_i,j+1− φn+1_i,j h , u v p u v p DXu p DX p u DXu p : p → u.

p u v p u v ∂/∂x u (DXu uu)i+1 2,j = u_i+3 2,j− ui−12,j 2h , (DXp uu)i,j = u_i+1 2,j− ui−12,j h Ω un+1 ∂Ω= 0 u∗ _φn+1

u∗ _{= u}n+1_{+ Δt∇φ}n+1_, u∗ ∂Ω= Δt∇φ n+1_ ∂Ω. qn+12 pn+12 u∗ u∗ _{= u}n+1_{+ O(h}2_), u∗ ∂Ω= u n+1_ ∂Ω = 0, ∇φn+1 ∂Ω· n = 0. Ω = [0, hN] × [0, hN] N× N h (i, j) [0, 1] × [0, 1] (i, j)  i +1 2  h,  j + 1 2  h  (1/2, j) (i, 1/2) (N +1/2, j) (i, N + 1/2)  ∂u∗ ∂x  3 2,j , u∗₅ 2,j− u ∗ 1 2,j 2h ,

...

∂/∂x 4 3h2  v(x + h, y) + 2v(x− h 2, y)− 3v(x, y)  = ∂2 ∂x2v(x, y) + O(h),  ∂v ∂x  1,j+1 2 = v2,j+12 + 3v1,j+12 − 4v12,j+12 3h + O(h2),  ∂2v ∂x2  1,j+1 2 = 4 3h2  v_2,j+1 2 + 2v12,j+12 − 3v1,j+12  + O(h). (1, j+1/2) (2, j+1/2) v v (1/2, j + 1/2) un _u∗ φn+1 ∂2φ(x, y) ∂x2 = ∂φ(x + h₂, y)/∂x− ∂φ(x − h₂, y)/∂x h + O(h 2_). x + h/2 φ(x + h, y)− φ(x, y) h2 − 1 h ∂φ(x− h₂, y) ∂x + O(h).

 ∂2 ∂x2φ  1,j = φ2,j − φ1,j h2 − 1 h  ∂ ∂xφ  1 2,j + O(h). (∂φ/∂x)1 2,j = 0 (1, j) (2, j) _p φ  ∂2φ ∂x2  1,j = φ2,j− φ1,j h2 ∇pn+1 2 u∗ v∗ u v DXu p DYvp n + 1 uk_{, φ}k_{, p}k−1 2, (u · ∇u)k+12 _{, (}∇q)n+12 _{, (Δ}u)k k ≤ n un+1 _pn+1 2 u∗ u∗_{− u}n Δt + (u · ∇u)n+ 1 2 + ∇qn+12 = μ 2(Δu∗+ (Δu) n_{) + Gn+}1 2  ₁ ΔtIuu− μ 2 Δuu  u∗ = −pn−_x 12 − (u · ∇u)n+12 + u n Δt + μ 2 (Δu) n_{+ G}n+1 2 1 ;  ₁ ΔtIvv− μ 2 Δvv  v∗ = −pn−_y 12 − (u · ∇v)n+12 + v n Δt+ μ 2(Δv) n_{+ G}n+1 2 2 ;

Iu u Ivv u v φn+1 ΔtΔφn+1 _{= ∇ · u}∗_. (∇ · u∗_{) = DX}p uu∗ + DYpvv∗; Δp pφn+1 = 1 Δt(∇ · u∗) ; k φn+1 φn+1+ k un+1 _{= u}∗_{+ Δt∇φ}n+1 un+1 = u∗− ΔtDXu_pφn+1; vn+1 = v∗− ΔtDYv_pφn+1; pn+12 = pn−12 + φn+1− μ 2 ∇ · u∗;

 ∂p ∂x n+1 2 = DXu ppn+ 1 2;  ∂p ∂y n+1 2 = DYv ppn+ 1 2; (Δu)n+1 _{= Δ}u uun+1; (Δv)n+1 _{= Δ}v vvn+1; (u · ∇u)n+32 = 3 2  un+1DXu_uun+1+ I_uvvn+1DYu_uun+1 −1₂(un_DXu uun+ IvuvnDYuuun) ; (u · ∇v)n+32 = 3 2  Iu vun+1DXvvvn+1+ vn+1DYvvvn+1  −1₂(Iu vunDXvvvn+ vnDYvvvn) ; Iv u Iuv (Iu vu)i,j+1 2 = u_i+1 2,j + ui−12,j + ui+12,j+1+ ui−12,j+1 4

u(x, y, t) =− sin2(πx) sin(2πy) cos t, v(x, y, t) = sin2(πy) sin(2πx) cos t, p(x, y, t) = μΔφ− ∂φ

∂t, φ

||E(u)||_∞ ||E(p)||_∞ ×10−02 _×10−02 ×10−03 _×10−03 ×10−04 _×10−03 ×10−04 _×10−03 ×10−05 _×10−04 ×10−05 _×10−04 ||E(u)||_∞ ||E(p)||_∞ ×10−02 ×10−03 ×10−03 ×10−04 G t = 10 μ = 1 N × N Δt = h = 1/N ∞ k

ordrek= log(E_log(Nk−1/Ek) k/Nk−1).

X(s)

s

Γ Ω

u = 0 Γ L s Γ X(s) : [0, L] → Ω s∈ Γ X ∈ Ω ˆs(s) X(s) X(0) X(ˆs) dˆs ds =  ddsX  . (₎ g(s) : Γ→ R dg dˆs = g  ₌ dg ds  d_dsX. τ n τ (s) = X_{(s) = Rn(s),} R π/2 Γ _X

Γ Ω s₁, s₂, ..., sp g gk := g(sk), k = 1, ..., p. ˜g(s) ∈ C2_{(Γ) L} ˜g(s) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˜g₀(s), s∈ [s₀, s₁], ˜g₁(s), s∈ [s₁, s₂], ... ˜g_p−1(s), s∈ [s_p−1, sp], ˜gp(s), s∈ [sp, s0]. j = 0, ..., p ˜gj ˜g(sj) = g(sj) d/ds ˜gj dg/ds C1(Γ) p

ˆ L Γ F (x) F (x) =  Lˆ 0 f(ˆs)δ(x − X(ˆs)) dˆs, f T (s) = T₀dX(s) ds   − 1, T₀ f f = d(T (s)τ (s)) ds  dX(s)ds  , (₎ f = (T (s)τ (s))_. ∂u ∂t + u · ∇u + ∇p = μΔu + G + F , F = 0 Ω+ Ω− u p Γ u, p ∈ C∞_{(Ω \ Γ).} G G ∈ C2_{(Ω \ Γ)}

[μu] 0 [μuτ] 0 [μun] −(f · τ )τ [μuττ] κ(f · τ )τ [μunτ] −((f · τ )τ ) [μunn] −κ(f · τ )τ + (f · τ )n + (f · n)τ − [G · τ ] τ [p] _{f · n} [p_τ] _{(f · n)} [p_n] _{[G · n] + (f · τ )} [u∗_{] [u]} [u∗ τ] [uτ] [u∗ n] [un] [u∗ ττ] [uττ] [u∗ nτ] [unτ] [μu∗ nn] [μunn] −μ2(u · n)(f · τ )τ u p μ κ(s) s g ∈ C(Ω \ Γ) g [g] (s) = lim ε→0+(g(X(s) + εn(s)) − g(X(s) − εn(s))) . u∗ _φn+1 φn+1 φn+1∈ C2[Ω] u∗

[u_x] = [u_n] n_x+ [u_τ] τ_x [u_xx] = [u_nn] n2 x+ 2 [unτ] τxnx+ [uττ] τx2, nx τx f : R → R x_i [x, x + h] M + 1 f : R → R f (x + h) = N  n=0 hn n!f (n)_{(x) + O(h}N+1_).

f : [x, x + h] → R f x₁ < x₂ <· · · < xM M + 1 h_m = x_m+1− x_m m = 0, ..., M (xM, x + h) f (x + h) = N  n=0 hn_M n! f (n)_(x+ M) + O(hN+1). [f]_M := lim ε→0+(f(xM + ε) − f(xM − ε)) . f xM f (x + h) = N  n=0 hn_M n!  f(n)(x−_M) +f(n)_M+ O(hN+1), f (x + h) = N  n=0 hn_M n! f (n)_(x− M) + N  n=0 hn_M n!  f(n)_M + O(hN+1). f(n)(x−_M) (x_M−1, xM) f (x + h) = N  n=0 hn_M n!  _N  k=n hk−n_M−1 (k − n)!f(k)(x+M−1)  + N  n=0 hn_M n!  f(n)_M + O(hN+1). N  n=0 hn_M n!  _N  k=n hk−n_M−1 (k − n)!f(k)(x+M−1)  = N  n=0 f(n)(x+_M−1)  _n  k=0 hk_M k! hn−k_M−1 (n − k)!  .

n!/n! N  n=0 f(n)(x+_M−1)  _n  k=0 hk_M k! hn−k_M−1 (n − k)!  = N  n=0 f(n)(x+_M−1) n!  _n  k=0 n! (n − k)!k!hkMhn−kM−1  = N  n=0 f(n)(x+_M−1) n! (hM + hM−1) n . f (x + h) = N  n=0 f(n)(x+_M−1) n! (hM + hM−1) n₊N n=0 hn_M n!  f(n)_M + O(hn+1). x_M−1 f (x+h) = N  n=0 f(n)(x−_M−1) n! (hM + hM−1) n₊N n=0 (x + h − xM−1)n n!  f(n) M−1+ N  n=0 hn_M n!  f(n)_M+ M + 1 h =M_m=0h_m f (x + h) = N  n=0 hn n!f (n)_{(x) +} M  m=1 N  n=0 (x + h − xm)n n!  f(n)_m+ O(hN+1). N

x x + h s₁, s₂, ..., sM X(s1) x X(s2) g ∈ C2(Ω \ Γ) Γ (x, x + h) X(s1), ..., X(sM) x [f](s₁) [f](s₂) g(x+h) = N  n=0 ||h||n n! g (n)_(x)+ M  m=1 (−1)m−1 N  n=0 ||x + h − xm||n n!  g(n)(sm)+O(||h||N+1) g(n) h (−1)m−1 x σ(x) σ(x) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 _{x ∈ Ω}+_, −1 x ∈ Ω−_.

g(x+h) = N  n=0 ||h||n n! g (n)_(x)+σ(x) M  m=1 (−1)m N  n=0 ||x + h − xm||n n!  g(n)(sm)+O(||h||N+1). C_gN(x, h) = σ(x) M  m=1 (−1)m N  n=0 ||hm||n n!  g(n)(sm). N g [x, x + h] h =||h|| g(x + h) = N  n=0 hn n!g (n)_{(x) + C}N g (x, h) + O(hN+1) g(x − h) = N  n=0 (−h)n n! g (n)_{(x) + C}N g (x, −h) + O(hN+1). g(x + h) − g(x − h) 2h = g(x) + 1 2h  C_g2(x, h) − C_g2(x, −h)+ O(h2), g(x + h) − 2g(x) + g(x − h) h2 = g _{(x) +} 1 h2  C_g2(x, h) + C_g2(x, −h)+ O(h). h (DXu uu)i+1 2,j =  ∂u ∂x  i+1 2,j + 1 2h  C_u2(x, hx) + C_u2(x, −hx)+ O(h2), hx = (h, 0)T 1 2h  C_u2(x, hx) + C_u2(x, −hx)

DXu uu C{DXu uu} DXu uu = ∂u ∂x + C{DX u uu} + O(h2). C{Δu uu} = 1 h2 

Cu(x, hx2 ) + Cu(x, −hx2 ) + C_u2(x, hy) + Cu(x, −hy2 )

DYu u C{DXu uun} C{DXvvvn} C{DXp uu} = 1 h  C_u2  x,h₂x  − C2 u  x,−h₂x  p C{DXu ppn+ 1 2} = 1 h  C_p1  x,hx 2  − C1 p  x,−hx 2  . ∂ku/∂tk ∂kp/∂tk k X(s) C{·} n + 1 uk_{, φ}k_{, p}k−1 2, (u · ∇u)k+12 _{, (}∇q)n+12 _{, (Δ}u)k

k ≤ n un+1 pn+12  ₁ ΔtIuu − μ 2 Δuu  u∗ = −pn−12 x − (u · ∇u)n+ 1 2 + u n Δt +μ 2 (Δu) n_{+ G}n+1 2 1 − μ 2 C{Δuuu∗};  ₁ ΔtIvv− μ 2 Δvv  v∗ = −pn−_y 12 − (u · ∇v)n+12 + v n Δt +μ 2(Δv) n_{+ G}n+1 2 2 − μ 2 C{Δvvv∗}; (∇ · u∗_{) = DX}p uu∗− C{DXpuu∗} + DYpvv∗− C{DYpvv∗}; Δp pφn+1= 1 Δt(∇ · u∗) + C{Δppφn+1}; un+1 = u∗ − ΔtDXu_pφn+1− C{DXu_pφn+1}; vn+1 = v∗− ΔtDYv_pφn+1− C{DYv_pφn+1}; pn+12 = pn−12 + φn+1−μ 2 (∇ · u∗) ; pn+_x 12 _{= DX}u ppn+ 1 2 − C{DXu_ppn+12}; pn+_y 12 _{= DY}v ppn+ 1 2 − C{DYv_ppn+12}; (Δu)n+1 _{= Δ}u uun+1− C{Δuuun+1}; (Δv)n+1_{= Δ}v vvn+1− C{Δvvvn+1};

(u · ∇u)n+32 = 3 2un+1  DXu uun+1− C{DXuuun+1}  +3 2Ivuvn+1  DYu uun+1− C{DYuuun+1}  −1₂(un_(DXu uun− C{DXuuun}) + Ivuvn(DYuuun− C{DYuuun})) ; (u · ∇v)n+32 = 3 2Iuvun+1  DXv vvn+1− C{DXvvvn+1}  +3 2vn+1  DYv vvn+1− C{DYvvvn+1}  −1₂(Iu vun(DXvvvn− C{DXvvvn}) + vn(DYvvvn− C{DYvvvn})) ; u Iv u Iuv g ∈ C2(Ω \ Γ) g(x + h/2) + g(x − h/2) 2 = g(x) + 1 2  C_g1(x, h/2) + C_g1(x, −h/2)+ O(h2). Ip u Ivp Iv u Iu v Ipv Iup Ω = [0, 1] × [0, 1]

Iv u Ipu Iv p μ = 0.1 (x − 0.5)2 0.352 + (y − 0.5)2 0.252 = 1. f(s) = −10n(s). F G = 0 Γ u = 0 ∇p = F . p [p] = f · n f · n = −10 −10 C p = ⎧ ⎪ ⎨ ⎪ ⎩ C x ∈ Ω+, C + 10 x ∈ Ω−.

||E(u)||_∞ ||E(p)||_∞ ×10−04 _×10−03 ×10−05 _×10−03 ×10−06 _×10−04 ×10−06 _×10−05 ||E(u)||_∞ ×10−15 _×10−16 _×10−16 ||E(v)||_∞ ×10−15 _×10−16 _×10−16 ||E(p)||_∞ ×10−15 _×10−15 _×10−15 N× N h = 1/N Δt = h 12×12

N = 24 N = 12

f(s, t) X(s, t) ∂X(s, t) ∂t = u(X(s, t), t). Γ t₀ + Δt t₀ X(s, t0+ Δt) = X(s, t0) +  _Δt 0 u(X(s, t0 + t), t₀+ t)dt.

 _Δt 0 u(X(s, t0 +t), t_{0+t)dt ≈ Δt}  u(X(s, t0), t0) + u(X(s, t0+ Δt), t0+ Δt) 2  . Γn Xn k = X(sk, nΔt) s₁, ..., sk Un k = u(Xkn, nΔt) Xn+1 k = Xkn+ ΔtU n k + Ukn+1 2 . Un+1 k Xkn+1 Un+1 k un+1

U

u Γ _u x0 = (x0, jh) j u Γn u x0 x1 = (x0−α, jh) x2 = (x0+ β, jh) α, β > 0

u(x1) = u(x0) − αux(x−₀) + O(h2),

u(x₀) u

C₁

u(x1) = u(x0) − αux(x+₀) + C₁+ O(h2).

u(x₁) = u(x₀) −α 2  ux(x+₀) + ux(x−₀)+ C1 2 + O(h2). u(x₂) u(x2) = u(x0) + β 2  ux(x+₀) + ux(x−₀)+C2 2 + O(h2), ⎛ ⎝ u(x1) u(x₂) ⎞ ⎠ = ⎛ ⎝ 1 −α 1 β ⎞ ⎠ ⎛ ⎝ u(x0) (u_x(x+ 0) + ux(x−0))/2 ⎞ ⎠ + 1 2 ⎛ ⎝ C1 C₂ ⎞ ⎠ + O(h2_). u(x₀) ⎛ ⎝ u(x0) (u_x(x+ 0) + ux(x−0))/2 ⎞ ⎠ = ⎛ ⎝ 1 −α 1 β ⎞ ⎠ −1⎛ ⎝ ⎛ ⎝ u(x1) u(x₂) ⎞ ⎠ − 1₂ ⎛ ⎝ C1 C₂ ⎞ ⎠ ⎞ ⎠+O(h2_). ⎛ ⎝ 1 −α 1 β ⎞ ⎠ −1 = 1 β + α ⎛ ⎝ β α −1 1 ⎞ ⎠ , u(x₀) = β(u(x1) − C1/2) + α(u(x2) − C2/2) α + β + O(h 2_).

u Γ u u(X(s, t), t) u(X(s, nΔt), nΔt) Γ u v Un_{(s) = u(X(s, nΔt), nΔt) + O(h}2_), Un k Un k = Un(sk) = u(X(sk, nΔt), nΔt) + O(h2). Γn+1 n Γn+1 0 Γn+1 Xn+1 _{= X}n_{+ ΔtU}n_{+ O(Δt}2_). Γn+1 0 Xn+1 0 = Xn+ ΔtUn. Γn+1 Γn+1 i Γn+1i+1 Γn+1 i Γn+1i+1 Xn+1 i+1 − Xin+1< ε ε Γn+1 i Γn+1i+1 Γn+1 i Γn+1i+1

i← 0 Γn+1 0 Γn+1 Xn+1 = Xn_{+ ΔtU}n C{} Γn+1 i un+1 Un+1 _un+1 Γn+1 i+1 Γn i← i + 1 Γn+1 i Γn+1i−1 (x − 0.5)2 a2 + (y − 0.5)2 b2 = 1, (u(x, 0) = 0) a = .35 + ε, b = .25 + ε ε f(s, t) = βκ(s, t)n(s, t), β

Δt = 1/32 16 × 16

p pn+12 = pn−12 + φn+1− μ 2 ∇ · u∗. [∇ · u∗_{] = O(Δt) φ}n+1 p v v∗ ∇ · u∗

u∗ u∗ _{≈ u}n+1 _{∇ · u}∗ u v p ∂un+12/∂t un+1_{− u}n Δt = ∂un+12 ∂t + O(Δt 2_), u ∂u/∂t ∂2u/∂t2 [nΔt, (n + 1)Δt]

(i, j) T T t₁, t₂, ..., tm Δt x [nΔt, (n + 1)Δt] _x Ω− _Ω+ Ω+ _Ω−

n+1 (i, j) T O(Δt) O(Δt) T [nΔt, (n + 1)Δt] (i, j) Γn _d0 x Γn+1 _d1 x u T = Δt  n + d 0 x d0_x+ d1_x  . O(Δt) T = Δt  n + d 0 y d0_y+ d1_y  + O(Δt). T = Δt  n + d 0 x/2 d0_x+ d1_x + d0_y/2 d0_y+ d1_y  + O(Δt). Γn _Γn+1

x = (x1, x2, (n + 1/2)Δt) O(Δt) (x, y, t) x = (x1, x2, (n+1/2)Δt) ht= (0, 0, Δt/2) g ∈ C2[Ω \ Γ] g(x + ht) = g(x) + C_g0(x, ht) + O(Δt), g(x − ht) = g(x) + C_g0(x, −ht) + O(Δt). [nΔt, (n + 1)Δt] C0 C_g0(x, ht) = ⎧ ⎨ ⎩ −σ(x − ht) [g] [(n + 1₂)Δt, (n + 1)Δt], 0 C_g0(x, −ht) = ⎧ ⎨ ⎩ σ(x − h_t) [g] [nΔt, (n + 1₂)Δt], 0 . [g]+_T C_g0(x, ht) [g]−_T C_g0(x, −ht) g(x + ht) = g(x) + [g]+_T + O(Δt), g(x − ht) = g(x) + [g]−_T + O(Δt).

g(x + ht) = g(x) +Δt 2 gt(x) + Cg1(x, ht) + O(Δt2), g(x − ht) = g(x) −Δt 2 gt(x) + Cg1(x, −ht) + O(Δt2). C_g1(x, ht) = ⎧ ⎪ ⎨ ⎪ ⎩ −σ(x) ([g] + ||ht|| [g]) T ∈ [(n +1₂)Δt, (n + 1)Δt), 0 C_g1(x, −h) = ⎧ ⎪ ⎨ ⎪ ⎩ −σ(x) ([g] − ||ht|| [g]) T ∈ [nΔt, (n + 1₂)Δt), 0 . ||ht|| = ⎧ ⎪ ⎨ ⎪ ⎩ (n + 1)Δt − T ∈ [nΔt, (n + 1 2)Δt), T − nΔt , σ(x) = ⎧ ⎪ ⎨ ⎪ ⎩ σ(x − ht) ∈ [nΔt, (n + 1₂)Δt), −σ(x − ht) . C_g1(x, ht) = ⎧ ⎪ ⎨ ⎪ ⎩ −σ(x − ht) ([g] + ((n + 1)Δt − T ) [gt]) T ∈ [(n + 1₂)Δt, (n + 1)Δt), 0 C_g1(x, −ht) = ⎧ ⎪ ⎨ ⎪ ⎩ σ(x − ht) ([g] + (T − nΔt) [gt]) T ∈ [(n +1₂)Δt, (n + 1)Δt), 0

DT ∂/∂t DT DTu uun+ 1 2 = u n+1_{− u}n Δt = ∂un+12 ∂t + C{DT u uun+ 1 2} + O(Δt). C{DTu uun+ 1 2} = 1 Δt  Cu(x, ht) − C1 _u1(x, −ht). C{DTv vvn+ 1 2} ut [ut] = −(u · n) [un] . Δμun+1 2 Δun+1 2 μ 2 

Δun+1_{+ Δu}n_{= μΔu}n+1

2 + μ 2  [Δu]+T + [Δu]−T  + O(Δt). n n + 1/2 (u · ∇u)n+12 (u · ∇u)n+1 2 = (u · ∇u)n+ O(Δt). n− 1 n

(u · ∇u)n+1 2 = ⎧ ⎨ ⎩ (u · ∇u)n _{[n − 1, n],} 3 2(u · ∇u)n−12(u · ∇u)n−1 . . n n + 1/2 C

C =− [u · ∇u]−_T = −u · [∇u]−_T .

pn+12 = pn−12 + O(Δt). (x₁, x₂) n− 1/2 n + 1/2 [(n − 1/2)Δt, (n + 1/2)Δt] [nΔt, (n + 1)Δt] ¯ T ∈ [(n − 1/2)Δt, (n + 1/2)Δt] ¯ x = (x1, x2, nΔt) [p]+_T¯ = C0 g( ¯x, ht) [p]−_T¯ = C0 g( ¯x, −ht). pn+12 = pn−12 +[p]+_T¯ − [p]−_T¯+ O(Δt). pn+12 = pn−12 + φn+1−μ 2(∇ · u∗), pn+12 = pn−12 +[p]_T+_¯ − [p]−_T¯+ φn+1− μ 2(∇ · u∗).

n+1 uk, φk, Δuk k ≤ n (u · ∇u)n+12  ₁ ΔtIuu− μ 2 Δuu  u∗ = −pn−_x 12 − (u · ∇u)n+12 + u · [∇u]− T +un Δt + μ 2 (Δu) n_{+ G}n+1₂ 1 − μ 2 C{Δuuu∗} + C{DTuuun} −μ₂ [Δu]+_T + [Δu]−_T;  ₁ ΔtIvv− μ 2 Δvv  v∗ = −pn−_y 12 − (u · ∇v)n+12 + u · [∇v]− T +vn Δt + μ 2 (Δv) n_{+ G}n+1 2 2 − μ 2 C{Δvvv∗} + C{DTvvvn} −μ₂ [Δv]+_T + [Δv]−_T; (∇ · u∗_{) = DX}p uu∗− C{DXpuu∗} + DYpvv∗− C{DYpvv∗}; Δp pφn+1= 1 Δt(∇ · u∗) + C{Δppφn+1}; un+1 = u∗ − ΔtDXu_pφn+1− C{DXu_pφn+1}; vn+1 = v∗− ΔtDYv_pφn+1− C{DYv_pφn+1}; pn+12 = pn−12 +[p]_T+_¯ − [p]−_T¯+ φn+1− μ 2 (∇ · u∗) ; pn+_x 12 _{= DX}u ppn+ 1 2 − C{DXu ppn+ 1 2}; pn+_y 12 _{= DY}v ppn+ 1 2 − C{DYv ppn+ 1 2};

(Δu)n+1 _{= Δ}u uun+1− C{Δuuun+1}; (Δv)n+1_{= Δ}v vvn+1− C{Δvvvn+1}; n n + 1 (u · ∇u)n+32 = 3 2un+1  DXu uun+1− C{DXuuun+1}  +3 2Ivuvn+1  DYu uun+1− C{DYuuun+1}  −1₂(un_(DXu uun− C{DXuuun}) + Ivuvn(DYuuun− C{DYuuun})) ; (u · ∇v)n+32 = 3 2Iuvun+1  DXv vvn+1− C{DXvvvn+1}  +3 2vn+1  DYv vvn+1− C{DYvvvn+1}  −1₂(Iu vun(DXvvvn− C{DXvvvn}) + vn(DYvvvn− C{DYvvvn})) ; n n + 1 (u · ∇u)n+32 = un+1DXu uun+1− C{DXuuun+1}  + Iv uvn+1  DYu uun+1− C{DYuuun+1}  ; (u · ∇v)n+32 = Iu vun+1  DXv vvn+1− C{DXvvvn+1}  + vn+1_DYv vvn+1− C{DYvvvn+1}  ;

u p

u∗ _φn+1

u u∗ _φn+1 φn+1 ∈ C2(Ω) u∗ un+1

Γ

Ω

Ω

Γ

Γ

[g] (s) = lim ε→0+(g(X(s) + εn(s)) − g(X(s) − εn(s))) , lim ε→0+  ∂Ωε g(X(s))dˆs =  Γ [g] (s)dˆs, ˆs _{X(s) X(0)} s ϕ ∈ C2(Ω) ∇ · Du Dt + Δp = ∇ · μΔu + ∇ · G + ∇ · F . μ ∇ · μΔu = μΔ(∇ · u) = 0. ϕ ∈ C2(Ω) Ωε lim ε→0+  Ωε  ∇ · Du Dt + Δp − ∇ · F − ∇ · G  ϕdx = 0. ϕΔp =∇ · (ϕ∇p) − ∇p · ∇ϕ = ∇ · (ϕ∇p) − ∇ · (p∇ϕ) + pΔϕ. lim ε→0+  Ωε ϕΔpdx = lim ε→0+  Ωε (∇ · (ϕ∇p − p∇ϕ) + pΔϕ) dx.

pΔϕ lim_ε→0+_Ω εpΔϕdV = 0 lim ε→0+  Ωε ϕΔpdx = lim ε→0+  ∂Ωε (ϕ∇p − p∇ϕ) · ndˆs. lim ε→0+  Ωε ϕΔpdx =  Γ [p_{n] ϕ − [p] ϕn}dˆs. G lim ε→0+  Ωε (∇ · G)ϕdx = lim ε→0+  Ωε (∇ · (ϕG) − G · ∇ϕ) dx. ϕ∈ C2(Ω) G · ∇ϕ lim ε→0+  Ωε G · ∇ϕdx = 0. G·∇ϕ lim ε→0+  Ωε (∇ · ϕG)dx = lim ε→0+  Ωε ϕG · ndˆs. lim ε→0+  Ωε (∇ · G)ϕdx =  Γ ϕ [G · n] dˆs. Du/Dt G lim ε→0+  Ωε ϕ∇ ·Du Dtdx =  Γ ϕ  Du Dt · n  dˆs. lim ε→0+  Ωε ϕ∇ · Du Dtdx = 0.

F  Ωε ϕ∇ · F dx =  Ωε ∇ · (ϕF )dx −  Ωε F · ∇ϕdx.  Ωε ∇ · (ϕF )dx =  ∂Ωε ϕF · ndˆs. ε > 0 0 F Γ  Ωε F · ∇ϕdx =  Ωε (F · n) ϕndx +  Ωε (F · τ ) ϕτdx. F F (x) =  Γf(ˆs)δ(x − X(ˆs))dˆs,  Ωε (F · n) ϕndx =  Ωε  Γf(s)δ(x − X(s))dˆs  · nϕndx =  Γ(f · n)ϕn dˆs.  Ωε (F · τ )ϕτ =  Γ(f · τ )ϕτ dˆs,  Γ(f · τ )ϕ _dˆs, (₎ ϕ(s) = dϕ(X(s)) dˆs  dX(s)_dˆs . Γ  Γ(f · τ )ϕ _{dˆs =−} Γ(f · τ ) _ϕdˆs. lim ε→0+  Ωε ϕ∇ · F dx =  Γ  (f · τ )ϕ− (f · n) ϕ_ndˆs

 Γ  [p_{n] − [G · n] − (f · τ )}ϕ + (f · n − [p]) ϕ_ndˆs = 0. ϕ(x) ∈ C2(Ω) ϕ(X(s)) ϕ_n(X(s)) C2(Γ) [p_{n] = [G · n] + (f · τ )}, [p] = f · n.

n

τ

η

ξ

χ(η)

(x, y)

Γ

Ω

Syst`eme de coordonn´ees local

Γ χ(η) p(x) dp dη = ∂p ∂η + χ _(η)∂p ∂ξ d2p dη2 = ∂2p ∂η2 + χ _(η)∂p ∂ξ + 2χ _(η) ∂2p ∂η∂ξ + (χ _(η))2 ∂2p ∂ξ2. χ(0) = χ(0) = 0 η = 0 κ = χ(0) dp dη   η=0 = ∂p ∂η   η=0 , d2p dη2   η=0 = ∂2p ∂η2   η=0 + κ∂p ∂ξ   η=0 . [p] _{= (f · n)} = [p_η] . p [p] = f · n, [p_{ξ] = [G · n] + (f · τ )}, [pη] = (f · n). u ϕ ∈ C2(Ω) Ωε lim ε→0+  Ωε  Du Dt + ∇p − μΔu − F − G  ϕdx = 0.

Du/Dt lim ε→0+  Ωε Du Dtϕdx = 0. G lim ε→0+  Ωε Gϕdx = 0. (∇p) ϕ = ∇ (pϕ) − p∇ϕ lim ε→0+  Ωε (∇p) ϕdx = lim ε→0+  Ωε ∇ (pϕ) dx − lim ε→0+  Ωε p∇ϕdx. p ∇ϕ 0 lim ε→0+  Ωε (∇p) ϕdx = lim ε→0+  Ωε ∇(pϕ)dx = lim ε→0+  ∂Ωε pϕndˆs =  Γ[p] nϕdˆs. ϕΔu ϕΔp lim ε→0+  Ωε ϕΔudx =  Γ[un] ϕ − [u] ϕn dˆs. F (x) =_Γδ(x − X(s))f(s)dˆs lim ε→0+  Ωε F (x)ϕdx =  Γf(s)ϕdˆs. ϕ∈ C2(Ω)  Γ(([p] n − [μun] − f(s)) ϕ + [μu] ϕn) dˆs = 0, [μu] = 0 [p] n − [μun] − f = 0. [p] = f · n − [μun] = f − (f · n) n, (ξ, η) [μuξ] = − (f · τ ) τ .

[μuξ] _{= − ((f · τ ) τ )} _{= [μuξη}] .

τ _{= κn}

[μuξη] = −(f · τ )_{τ − κ(f · τ )n.}

[μu] = 0 [μu] = 0 = [μuη]

[μu]= 0 = [μuηη] + κ [μuξ] . [μuξ] = − (f · τ ) τ [μuηη] = κ (f · τ ) τ . [μuξξ]  Du Dt  + [∇p] = [μΔu] + [F ] + [G] . [F ] = 0 F Du/Dt [μuξξ] [μuξξ] = − [μuηη] + [∇p] − [G] . [μuξξ] = −κ(f · τ )τ + (f · τ )_{n + (f · n)}_{τ − [G · τ ] τ .}

[ut] [u] = 0

D

Dt[u] = 0 = [ut] + u · [∇u] , u · [∇u] = (u · n) [uξ] [uη] = 0 [ut] = − (u · n) [uξ] .

u

Γn+1 _pn−1 2 Γn+1 pn−12 un+12 un un+1 p(x, (n − 1/2)Δt) = ⎧ ⎨ ⎩ p₊(x, (n − 1₂)Δt) Γn−12, p−(x, (n − 1₂)Δt) Γn−12, p₊, p₋ ∈ C∞(Ω) pn−12 n pn−12 = ⎧ ⎨ ⎩ p+(x, (n −1₂)Δt) Γn+1, p₋(x, (n −1₂)Δt) Γn+1.  pn−12  Γn+1  pn−12  =pn+1+ O(Δt). n + 1 fn+1 _f  μu∗_ξ= −(f · τ )τ , [μu∗_{] = 0.} [μuξ] _[μu] [μu∗_] _{= 0 =}_μu∗ η  , [μu∗_]_{= 0 =}_μu∗ ηη  + κμu∗_ξ,  μu∗_ξ = ((−f · τ )τ ) =μu∗_ξη.

n Γn+1 2  μu∗_η= 0,  μu∗_ηη= κ(f · τ )τ ,  μu∗_ξη= ((−f · τ )τ ).  u∗ ξξ   u∗_{− u}n Δt 

+ u · [∇u] + [∇p] =μ₂(Δu + Δu∗_{)+ [F ] + [G] .}

 u∗_{+ u} Δt  = 0 [F ] = 0, F u · [∇u] = (u · τ ) [uη] + (u · n) [uξ] = (u · n) [uξ]

[uη] = 0.  μu∗_ξξ  μu∗_ξξ = −μu∗_ηη− [μuηη] − [μuξξ] + 2 (u · n) [u_n] + 2 [∇p] − 2 [G] . [μuξξ] = − [μuηη] + [∇p] − [G] ,  μu∗_ξξ = − [∇p] + [G] −μu∗_ηη+ 2 (u · n) [uξ] + 2 [∇p] − 2 [G] ,  μu∗_ηη= κ(f ·τ )τ [μuξ] = −(f ·τ )τ  μu∗_ξξ =  −2u · n μ − κ  (f · τ )τ + [∇p] − [G] .  μu∗_ξξ= [μuξξ] − 1 μ(2u · n)(f · τ )τ . u∗ _un+1

φ

 φn+1_ξη  = 0  φn+1_ηη = 0. [∇ · u∗_{] = [∇ · u] + Δt}_Δφn+1_, ∇ · u = 0 u∗ η  = 0 ΔtΔφn+1₌_u∗ ξ  · n.  φn+1_ηη = 0  φn+1_ξξ = 1 Δt  u∗ ξ  · n.  u∗ ξ   u∗ ξ  · n = 0  φn+1_ξξ = 0. [p] = [p] +φn+1− Δt 2 μ  Δφn+1_.  φn+1 = 1 2  μu∗_ξ· n = 0. φn+1 ∈ C2(Ω)