発展方程式の数値解析 ii. keller-segel の走化性モデ …発展方程式の数値解析...
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発展方程式の数値解析II. Keller-Segelの走化性モデルと有限体積法
齊藤 宣一 (Norikazu SAITO)
東京大学大学院数理科学研究科
2011年 8月 27日発展方程式若手セミナー
http://www.infsup.jp/saito/materials/110827tsukuba2.pdf
N. Saito (UT) Evolution Equations II August 27, 2011 1 / 32
目次
1 Keller-Segel system modeling chemotaxis
2 A remark on the positivity of a solution of a convection-diffusion equation
3 Introduction to the finite volume method
4 Numerical examples
N. Saito (UT) Evolution Equations II August 27, 2011 2 / 32
Keller-Segel の走化性モデル
E. F. Keller & L. A. Segel : J. Theor. Biol. 26 (1970)
Mathematical models to describe the aggregation of cellular slime molds which move preferentially to relatively
high concentrations of a chemical secreted by themselves
u = u(x, t): the cell density of the slime molds;
v = v(x, t): the concentration of the chemical substance;
∇ϕ(v): the velocity of u due to chemotaxis.
N. Saito (UT) Evolution Equations II August 27, 2011 4 / 32
Keller-Segel の走化性モデルE. F. Keller & L. A. Segel : J. Theor. Biol. 26 (1970)
Mathematical models to describe the aggregation of cellular slime molds which move preferentially to relatively
high concentrations of a chemical secreted by themselves
u = u(x, t): the cell density of the slime molds;
v = v(x, t): the concentration of the chemical substance;
∇ϕ(v): the velocity of u due to chemotaxis.
u
Diffusion
N. Saito (UT) Evolution Equations II August 27, 2011 4 / 32
Keller-Segel の走化性モデルE. F. Keller & L. A. Segel : J. Theor. Biol. 26 (1970)
Mathematical models to describe the aggregation of cellular slime molds which move preferentially to relatively
high concentrations of a chemical secreted by themselves
u = u(x, t): the cell density of the slime molds;
v = v(x, t): the concentration of the chemical substance;
∇ϕ(v): the velocity of u due to chemotaxis.
f(v)
u
uu
Chemotaxis
N. Saito (UT) Evolution Equations II August 27, 2011 4 / 32
Keller-Segel の走化性モデルE. F. Keller & L. A. Segel : J. Theor. Biol. 26 (1970)
Mathematical models to describe the aggregation of cellular slime molds which move preferentially to relatively
high concentrations of a chemical secreted by themselves
u = u(x, t): the cell density of the slime molds;
v = v(x, t): the concentration of the chemical substance;
∇ϕ(v): the velocity of u due to chemotaxis.
u v
f(v)
Creation of field
N. Saito (UT) Evolution Equations II August 27, 2011 4 / 32
Diffusion Chemotaxis Creation of field
Rem: Flux of u is given by j = −Du∇u + u∇ϕ(v).
Keller-Segel system:
ut = −∇ · j = ∇ · (Du∇u − u∇ϕ(v))
kvt = Dv∆v + g(u, v)
Boundary conditions:
−j · ν = Du
∂u
∂ν− u
∂ϕ(v)
∂ν= 0
∂v
∂ν= 0.
N. Saito (UT) Evolution Equations II August 27, 2011 5 / 32
Keller-Segel system (E. F. Keller & L. A. Segel 1970)
ut = ∇ · (Du∇u − u∇ϕ(v)) in Ω × (0, T ),
kvt = Dv∆v − k1v + k2u in Ω × (0, T ),
∂u
∂ν= 0,
∂v
∂ν= 0 on ∂Ω × (0, T ),
u|t=0 = u0, v|t=0 = v0 on Ω
(KS)
Ω: bounded domain in Rd (d = 2, 3)
Mathematical model for aggregation phenomenon of slime molds resultingfrom their chemotactic features
u density of the cellular slime molds;v concentration of the chemical substance secreted by molds;Du, Dv diffusion coefficients, k relaxation time (≥ 0) and k1v − k2u ratio ofgeneration/extinction.ϕ(v) sensitive function (ϕ : R → R smooth and non-deceasing),
Conservation laws: positivity and total mass; ‖u(t)‖L1 = ‖u0‖L1 .
The solution may blow up. (It depends on ‖u0‖L1 and d.)
N. Saito (UT) Evolution Equations II August 27, 2011 6 / 32
Well-posedness
Ω and u0(x) are “smooth”
⇒ ∃(u, v): unique classical solution on [0, T ], T ∈ (0, Tmax).
Tmax: the supremum of the existence time;
Tmax = ∞ ⇒ global existence;Tmax < ∞ ⇒ blow up (in finite time) i.e., lim
t→Tmax
‖u(t)‖L∞ = +∞.
Horstmann (survey, 2003, 2004), Suzuki & Senba (book, 2004), Suzuki(book, 2005), ......
Example. Assume k = 0.
Ω ⊂ R1 ⇒ Tmax = +∞.
Ω ⊂ R2,
(i) ‖u0‖L1(Ω) < 4πλ−1 ⇒ Tmax = ∞;
(ii) ‖u0‖L1(Ω) > 8πλ−1 and
∫
Ω
|x − x0|2u0(x) dx 1 for some x0 ∈ Ω. ⇒
Tmax < ∞.
N. Saito (UT) Evolution Equations II August 27, 2011 7 / 32
Motivation
1 uが (凝集による)集中化を相当におこしても,安定に計算を遂行できる数値計算スキームの提案
2 L1 ノルムの保存や自由エネルギーの保存を (できるだけ)再現3 理論的な裏付け (安定性・収束解析,陽的な誤差評価)
注意:
E. Nakaguchi and Y. Yagi (Hokkaido Math. J. 2002), A. Chertock and A.Kurganov (Numer. Math. 2008), Y. Epshteyn and A. Izmirlioglu (J. Sci. Comput.2009), (SIAM J. Numer. Anal. 2008/09)
J. Haskovec and C. Schmeiser (J. Stat. Phys. 2009).
F. Filbet (Numer. Math. 2006) finite volume
S (IMA J. Numer. Anal. 2007), (RIMS Bessatsu 2009), (CPAA 2012)
conservative finite element
N. Saito (UT) Evolution Equations II August 27, 2011 8 / 32
§2. A remark on the positivity of a numericalsolution of a convection-diffusion equation
N. Saito (UT) Evolution Equations II August 27, 2011 9 / 32
Illustration of the issue
Model problem: u(x, t), b(x, t) ≥ 0: 1-periodic functions in x
ut = uxx − (b(x, t)u)x (x ∈ [0, 1), 0 < t < T )
Positivity u(x, 0) ≥ 0, 6≡ 0 ⇒ u(x, t) > 0 (t > 0)
Finite difference method:
xi = ih (h = 1/N),tn = τ1 + τ2 + · · · + τn: Grid points;
bni = b(xi, tn);
uni ≈ u(xi, tn): finite difference approximation.
forward Euler = central difference + central difference
uni − un−1
i
τn
=un−1
i−1 − 2un−1i + un−1
i+1
h2−
bn−1i+1 un−1
i+1 − bn−1i−1 un−1
i−1
2h
N. Saito (UT) Evolution Equations II August 27, 2011 10 / 32
Numerical example
b(x, t) = 4(1 + cos 2πx)(1 + t)2,h = 2−5,τj = 0.4 · h2
0
0.2
0.4
0.6
0.8
0 0.5
1 1.5
2 2.5
3 3.5
4
0
2
4
0 ≤ tn ≤ 4.0
N. Saito (UT) Evolution Equations II August 27, 2011 11 / 32
Numerical example
b(x, t) = 4(1 + cos 2πx)(1 + t)2,h = 2−5,τj = 0.4 · h2
0
0.2
0.4
0.6
0.8
1 0 0.5
1 1.5
2 2.5
3 3.5
4 4.5
-8
-4
0
4
8
0 ≤ tn ≤ 4.4
N. Saito (UT) Evolution Equations II August 27, 2011 11 / 32
Numerical example
b(x, t) = 4(1 + cos 2πx)(1 + t)2,h = 2−5,τj = 0.4 · h2
0
0.2
0.4
0.6
0.8
1 4.37 4.375
4.38 4.385
4.39 4.395
4.4
-80
-40
0
40
80
4.37 ≤ tn ≤ 4.4
N. Saito (UT) Evolution Equations II August 27, 2011 11 / 32
Numerical example
b(x, t) = 4(1 + cos 2πx)(1 + t)2,h = 2−5,τj = 0.4 · h2
-50
-40
-30
-20
-10
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1
u-ax
is
x-axis
4.37 ≤ tn ≤ 4.4 (another view-point)
N. Saito (UT) Evolution Equations II August 27, 2011 11 / 32
Consideration
Finite difference scheme(
λn = τn/h2)
⇔
uni = (1 − 2λn)un−1
i +(
λn +τn
2hbn−1i−1
)
un−1i−1 +
(
λn −τn
2hbn−1i+1
)
un−1i+1
Non-negativity (*) un−1i ≥ 0 (∀i) ⇒ un
i ≥ 0 (∀i)
A sufficient condition:
(· · · ) ≥ 0,(· · · ) ≥ 0,(· · · ) ≥ 0 ⇒ (*)
h ≤1
2βn−1⇒ (*).
(
βn = max1≤i≤N
bni
)
Before computation, we have to choose h satisfying:
h ≤1
2βT
, βT = max0≤tn≤T
βn.
N. Saito (UT) Evolution Equations II August 27, 2011 12 / 32
Upwind finite difference scheme
forward Euler = central difference+ upwind difference
uni − un−1
i
τn
=un−1
i−1 − 2un−1i + un−1
i+1
h2−
bn−1i un−1
i − bn−1i−1 un−1
i−1
h
⇔
uni =
(
1 − 2λn −τn
hbn−1i
)
un−1i +
(
λn +τn
hbn−1i−1
)
un−1i−1 + λnun−1
i+1
A sufficient condition: τn ≤h2
2 + hβn−1⇒ (*).
In each time step, we re-choose τn to satisfy the above condition.
Extension to d ≥ 2 and arbitrary Ω
FEM; Tabata (1977), Heinrich et al. (1977) → flow problems.In general, the upwind FEM destroys the conservation of mass.
Conservative numerical schemes;
FEM; Baba-Tabata upwinding (1981),Finite volume method (FVM), .....
N. Saito (UT) Evolution Equations II August 27, 2011 13 / 32
§3. Introduction to the finite volume method
N. Saito (UT) Evolution Equations II August 27, 2011 14 / 32
Convection-diffusion problem (model problem)
Find a function u = u(x, t) of (x, t) ∈ Ω × [0, T ] satisfying
ut −∇ · (∇u − bu) = f in Ω × (0, T ),
u = g on ∂Ω × (0, T ),
u|t=0 = u0 on Ω,
(1)
Ω: bounded polygonal/smooth domain in R2
Remark: It is straightforward to extend the results to the 3D case; we needsome trivial modifications.
T : positive constant
f : Ω × (0, T ) → R, g : ∂Ω × (0, T ) → R, u0 : Ω → R: smooth
b : Ω × (0, T ) → R2 smooth
N. Saito (UT) Evolution Equations II August 27, 2011 15 / 32
Admissible mesh: definition [Eymard et al. 2000]
D = Dii∈Λ: admissible mesh ⇔
(A1) Di: (open) convex polygonal domain; Ω = ∪Di | i ∈ Λ
(A2) Di ∩ Dj = entire common side σij / vertex / empty set (i 6= j)
(A3) ∃Pii∈Λ such that Pi ∈ Di and that the line segment connecting Pi with Pj
is orthogonal to the line including σij , if Di and Dj share a common side σij .
(A4) ∃ side σ of Di: σ ⊂ ∂Ω ⇒ Pi ∈ σ.
h = hD = maxdiam (Di) | i ∈ Λ
Di: control volume
Λ = Λ ∪ ∂Λ,Λ: index set of interior control volumes∂Λ: index set of boundary control volumes
Pi
jP
i js
Di
Dj
ni j
N. Saito (UT) Evolution Equations II August 27, 2011 16 / 32
Admissible mesh: examples (acute triangulations)
iP
PC
circumcentric domain(PC = circumcenter)
iP
GP
barycentric domain(PG = barycenter)
N. Saito (UT) Evolution Equations II August 27, 2011 17 / 32
Admissible mesh: examples (acute triangulations)
acute triangulation admissible mesh
Lemma: The circumcentric dual mesh of an acute triangulation is an admissiblemesh.
Remark: The barycentric dual mesh does not imply an admissible mesh ingeneral.
N. Saito (UT) Evolution Equations II August 27, 2011 17 / 32
Admissible mesh: examples (acute triangulations)
acute triangulation admissible mesh
Lemma: The circumcentric dual mesh of an acute triangulation is an admissiblemesh.
Remark: The barycentric dual mesh does not imply an admissible mesh ingeneral.
N. Saito (UT) Evolution Equations II August 27, 2011 17 / 32
Admissible mesh: examples (Voronoi diagram)
Voronoi diagram Voronoi diagram
N. Saito (UT) Evolution Equations II August 27, 2011 18 / 32
Admissible mesh: examples (Voronoi diagram)
admissible mesh admissible mesh
Lemma: A Voronoi diagram implies an admissible mesh if the number of pointsPi which are located on ∂Ω is large enough.
N. Saito (UT) Evolution Equations II August 27, 2011 18 / 32
Definitions
Piecewise constant functions:
Xh = vh ∈ L∞(Ω) | vh|Di= Const. (i ∈ Λ)
Vh = vh ∈ Xh | vh|Di= 0 (i ∈ ∂Λ)
τij =mij
dij
transmissibility
mij = length of σij
dij = |Pi − Pj |
mi = area of Di
Pi
jP
i js
Di
Dj
ni j
N. Saito (UT) Evolution Equations II August 27, 2011 19 / 32
Approximation of the diffusion part
∫
Di
(−∆u) dx = −
∫
∂Di
∇u · νi dS
= −∑
j∈Λi
∫
σij
∇u · νij dS
≈ −∑
j∈Λi
∫
σij
u(Pj) − u(Pi)
dij
dS
= −∑
j∈Λi
τij(u(Pj) − u(Pi)).
P5
7P
5,7s
D5
D7
D2
D4
5,2s
5,4s
τij =mij
dij
, mij = length of σij , dij = |Pi − Pj |
Λi = j ∈ Λ | Pi and Pj share the entire common side σij
(Λ5 = 2, 4, 7)
N. Saito (UT) Evolution Equations II August 27, 2011 20 / 32
Approximation of the convection part
∫
Di
∇ · (bu) dx =∑
j∈Λi
∫
σij
u (b · νi) dS
≈∑
j∈Λi
∫
σij
[(1 − rij)u(Pj) + riju(Pi)] (b · νi) dS
=∑
j∈Λi
[(1 − rij)u(Pj) + riju(Pi)] βij .
0 ≤ rij ≤ 1: the weighting parameter
rij =
1 (βij ≥ 0)
0 (βij < 0)
βij =
∫
σij
b · νij dS
P5
7P
5,7s
D5
D7
D2
D4
5,2s
5,4s
N. Saito (UT) Evolution Equations II August 27, 2011 21 / 32
Finite volume scheme
Find unh
nn=0 ⊂ Xh satisfying
uni − un−1
i
∆tmi −
∑
j∈Λi
τij(unj − un
i )
+∑
j∈Λi
βnij
[
(1 − rnij)u
nj + rn
ijuni
]
= fni mi
(i ∈ Λ, 1 ≤ n ≤ n),
uni = gn
i (i ∈ ∂Λ, 1 ≤ n ≤ n), u0i = u0,i (i ∈ Λ),
(2)
uni = un
h(Pi), tn = n∆t, ∆t = T/n.
fni = (1/mi)
∫
Di
f(x, tn) dx or fni = f(Pi, tn).
gni and u0,i are defined similarly.
N. Saito (UT) Evolution Equations II August 27, 2011 22 / 32
Variational expression
Find unh
nn=0 ⊂ Xh satisfying
(
unh − un−1
h
∆t, vh
)
h
+ ah(unh, vh) + bn
h(unh , vh) = (fh, vh)h
(∀vh ∈ Vh),
unh − gn
h ∈ Vh, u0h = u0h
(2)
(uh, vh)h =∑
i∈Λ
uivimi (uh ∈ Xh, vh ∈ Vh)
ah(uh, vh) = −∑
i∈Λ
vi
∑
i∈Λj
τij(uj − ui)
bnh(uh, vh) =
∑
i∈Λ
vi
∑
i∈Λj
[(1 − rnij)u
nj + rn
ijui]βnij
N. Saito (UT) Evolution Equations II August 27, 2011 23 / 32
Keller-Segel system
In Ω = (0, 1)2, consider the Keller-Segel system:
ut = ∇ · (∇u − u∇ϕ(v)) in Ω × (0, T ),
kvt = ∆v − 0.1v + 0.01u in Ω × (0, T ),
∂u
∂ν= 0,
∂v
∂ν= 0 on ∂Ω × (0, T ),
u|t=0 = u0 v|t=0 = v0 on Ω.
Space variable is discretized by a conservative finite element approximation inthe acute triangulation
Time variable is discretized by the forward Euler with a time incrementcontrol of S (2007, 2009, 2012).
N. Saito (UT) Evolution Equations II August 27, 2011 25 / 32
Keller-Segel system
unh =
unh
‖unh‖∞
の等高線
φ(v) = 5v(1) k = 0.1 (T = 0.19)b0214-43u.wmv; b0214-43v.wmv(2) k = 0.001 (T = 0.14)b0214-46u.wmv; b0214-46v.wmv
φ(v) = 20 log v(3) k = 0.1 (T = 0.21)b0214-57u.wmv; b0214-57v.wmv(4) k = 0.001 (T = 0.27)b0214-58u.wmv; b0214-58v.wmv
φ(v) = 5v2
(5) k = 0.1 (T = 0.011)b0214-63u.wmv; b0214-63v.wmv(6) k = 0.001 (T = 0.002)b0214-66u.wmv; b0214-66v.wmv
***u.wmvが uh
***v.wmvが vh
‖u0h‖1 = 50.0
Windows Media Player で再生して下さい.Macの方は,Flip 4 Macなどをインストールして下さい.
N. Saito (UT) Evolution Equations II August 27, 2011 26 / 32
Degenerate Keller-Segel system
In a bounded domain Ω, consider a degenerate Keller-Segel system:
ut = ∇ ·(
∇um − k0u∇vl)
in Ω × (0, T ),
0 = ∆v − v + u in Ω × (0, T ),
∂
∂νum − k0u
∂
∂νvl = 0,
∂v
∂ν= 0 on ∂Ω × (0, T ),
u|t=0 = u0 on Ω.
Space variable is discretized by the cell-centered finite volume approximationin the admissible mesh.
Time variable is discretized by the forward Euler with a time incrementcontrol of S (2007, 2009, 2012).
N. Saito (UT) Evolution Equations II August 27, 2011 27 / 32
Numerical example (A): Ω = (0, 1)2
Webブラウザー (Internet Explorer, Netscape, Safariなど)で開いて下さい.m ‖u0‖L1
d0209-1.html 1.0 20.0d0209-2.html 1.2 20.0d0209-3.html 1.4 20.0
Ω = (0, 1)2 square
f(u) = um, g(v) = k0v
k1 = k2 = 1.0, k0 = 5.0
‖u0h‖L1 = 20 > 4π > 2π
If m = 1, the boundary/cornerblow up may occur.
0
0.2
0.4
0.6
0.8
1 0 0.2
0.4 0.6
0.8 1
0
0.2
0.4
0.6
0.8
1
u0h
left figure right figure
unh un
h/‖unh‖L∞
N. Saito (UT) Evolution Equations II August 27, 2011 28 / 32
Numerical example (A’): Ω = (0, 1)2
md0209-1x.html 1.0d0209-2x.html 2.0d0209-3x.html 3.0
Ω = (0, 1)2 square
f(u) = um, g(v) = k0v
k1 = k2 = 1.0, k0 = 5.0
x=a; ‖u0‖L1 = 1.0
x=b; ‖u0‖L1 = 5.0
x=c; ‖u0‖L1 = 10.0
x=d; ‖u0‖L1 = 20.0
0
0.2
0.4
0.6
0.8
1 0 0.2
0.4 0.6
0.8 1
0
0.2
0.4
0.6
0.8
1
u0h
left figure right figure
unh un
h/‖unh‖L∞
N. Saito (UT) Evolution Equations II August 27, 2011 29 / 32
Numerical example (B): Ω = B(0, 1)
m ‖u0‖L1
d0221-1.html 1.0 20.0d0221-2.html 1.2 20.0d0221-3.html 1.4 20.0
Ω = B(0, 1) disk
f(u) = um, g(v) = k0v
k1 = k2 = 1.0, k0 = 2.0
The smooth boundary is exactlydiscretized.
-1
-0.5
0
0.5
1 -1
-0.5
0
0.5
1 0
0.2
0.4
0.6
0.8
1
left figure right figure
unh un
h/‖unh‖L∞
N. Saito (UT) Evolution Equations II August 27, 2011 30 / 32
Numerical example (C): Ω = B(0, 1)
m l ‖u0‖L1
d0225-6 8.html 1.0 1.0 80.0d0225-6 8.html 1.0 2.0 80.0d0301-6 8.html 1.2 1.0 80.0d0301-6 8.html 1.2 2.0 80.0
Ω = B(0, 1) disk
f(u) = um, g(v) = k0vl
k1 = k2 = 1.0, k0 = 1.0
The smooth boundary is exactlydiscretized.
-1
-0.5
0
0.5
1 -1
-0.5
0
0.5
1 0
0.2
0.4
0.6
0.8
1
left figure right figure
unh un
h/‖unh‖L∞
N. Saito (UT) Evolution Equations II August 27, 2011 31 / 32
Thank you for your attention!
Norikazu SAITO
Graduate School of Mathematical Sciences
The University of Tokyo
http://www.infsup.jp/saito/
N. Saito (UT) Evolution Equations II August 27, 2011 32 / 32