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1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Lab session: numerical simulations of sponateouspolarization
Emeric Bouin & Vincent CalvezCNRS, ENS Lyon, France
CIMPA, Hammamet, March 2012
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Spontaneous cell polarization: the 1D case
The Hawkins-Voituriez model for spontaneous polarization in 1Dreads as follows:
∂tρ(t, x) = ∂xxρ(t, x) + ρ(t, 0)∂xρ(t, x) , t > 0 , x ∈ (0,+∞) .
The equation is a transport-diffusion equation.
Numerical issues:
• blow-up,
• non-trivial steady states,
• self-similar decay.
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Behaviour of solutions
TheoremThere is a nice and simple dichotomy:
• If M < 1, the solution is global in time. It converges towardsa (unique) self-similar profile GM :
limt→+∞
∥∥∥∥ρ(t, x)− 1√t
GM
(x√t
)∥∥∥∥L1
= 0 .
• If M = 1 there is a one-parameter family of steady states:να(x) = α exp(−αx).Convergence holds + the first moment is conserved:α−1 =
∫x>0 ρ
0(x) dx.
• If M > 1 and ∀x > 0 ∂xρ0(x) ≤ 0, the solution blows up in
finite time.
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Contents
1- Diffusion equation
2- The diffusion-transport equation
3- The diffusion-transport equation in self-similar variables
4- Cluster formation
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Contents
1- Diffusion equation
2- The diffusion-transport equation
3- The diffusion-transport equation in self-similar variables
4- Cluster formation
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
The diffusion equation in the half-line
We begin with the diffusion equation
∂tρ(t, x) = ∂xxρ(t, x) , x > 0
+ Neumann boundary condition at x = 0
We rewrite the equation in ”divergence” form,
∂tρ(t, x) + ∂xF = 0 ,
The flux F is given by Fick’s law: F = −∂xρ.
The boundary condition at {x = 0} is F (0) = 0 (no-flux boundarycondition).
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Time-Space discretization
We discretize the function ρ(t, x) on a regular grid
[0 : ∆t : T ]× [0 : ∆x : L] ,
• ∆t is the time step,
• T is the final time of computation,
• ∆x is the space step,
• L is the length of the numerical interval.
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Strategy for computing ρ(t, x) numerically
We replace the equation ∂tρ(t, x) + ∂xF = 0 with the numericaldiscretization
ρ(t + ∆t, x)− ρ(t, x)
∆t+
F (t, x + ∆x/2)− F (t, x −∆x/2)
∆x= 0 .
(1)We introduce
• ρni = ρ(n∆t, i∆x)
• F ni± 1
2
= F (n∆t, x ±∆x/2).
The equation (1) rewrites at (t, x) = (tn, xi ),
ρn+1i − ρn
i
∆t+
F ni+ 1
2
− F ni− 1
2
∆x= 0
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Boundary conditions
The boundary condition F (0) = 0 reads
F 12
= 0 .
For i = 1 the equation (1) reads
ρn+11 − ρn
1
∆t+
F n1+ 1
2
− 0
∆x= 0
We have similarly for i = Nx
ρn+1Nx − ρ
nNx
∆t+
0− F nNx− 1
2
∆x= 0
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
The diffusion equation
We have F = −∂xρ, therefore
F (t, x + ∆x/2) = −ρ(t, x + ∆x)− ρ(t, x)
∆x
The scheme writes finally
ρn+1i − ρn
i
∆t+
1
∆x
(−ρni+1 − ρn
i
∆x+ρni − ρn
i−1
∆x
)= 0 ,
Remark. It coincides with the classical finite difference scheme forthe heat equation
ρn+1i =
(1− 2
∆t
∆x2
)ρni +
∆t
∆x2ρni+1 +
∆t
∆x2ρni−1 ,
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Explicit scheme for the diffusion equation
• Left boundary condition (zero-flux) when i = 1
ρn+1i − ρn
i
∆t+
1
∆x
(−ρni+1 − ρn
i
∆x+ 0
)= 0 ,
• Right boundary condition (zero-flux) when i = Nx
ρn+1i − ρn
i
∆t+
1
∆x
(0 +
ρni − ρn
i−1
∆x
)= 0 ,
• Flux of the density when 2 ≤ i ≤ Nx − 1
ρn+1i − ρn
i
∆t+
1
∆x
(−ρni+1 − ρn
i
∆x+ρni − ρn
i−1
∆x
)= 0 ,
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Explicit scheme for the diffusion equation
• Left boundary condition (zero-flux) when i = 1
ρn+1i − ρn
i
∆t+
1
∆x
(−ρni+1 − ρn
i
∆x+ 0
)= 0 ,
• Right boundary condition (zero-flux) when i = Nx
ρn+1i − ρn
i
∆t+
1
∆x
(0 +
ρni − ρn
i−1
∆x
)= 0 ,
• Flux of the density when 2 ≤ i ≤ Nx − 1
ρn+1i − ρn
i
∆t+
1
∆x
(−ρni+1 − ρn
i
∆x+ρni − ρn
i−1
∆x
)= 0 ,
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Explicit scheme for the diffusion equation
• Left boundary condition (zero-flux) when i = 1
ρn+1i − ρn
i
∆t+
1
∆x
(−ρni+1 − ρn
i
∆x+ 0
)= 0 ,
• Right boundary condition (zero-flux) when i = Nx
ρn+1i − ρn
i
∆t+
1
∆x
(0 +
ρni − ρn
i−1
∆x
)= 0 ,
• Flux of the density when 2 ≤ i ≤ Nx − 1
ρn+1i − ρn
i
∆t+
1
∆x
(−ρni+1 − ρn
i
∆x+ρni − ρn
i−1
∆x
)= 0 ,
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Structure of the scilab code 1/4
Space and time grid of resolution.
%% Space discretisation
L = 10;dx = 0.1;x = [0:dx:L];Nx = length(x);
%% Time discretisation
T = 10;dt = dx^2/4;t = [0:dt:T];Nt = length(t);
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Structure of the scilab code 2/4
Initial data ρ0: we start with a gaussian
%% Initial condition for the density of molecules rho_0
sigma = 1;rho0 = exp(-x.^2/sigma^2);
Z = sum(rho0*dx);rho0 = (M/Z)*rho0;
rho1 = rho0;
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Structure of the scilab code 3/4
Time loop to update the value of the density ρ at each time step n
for n = 1:Nt
rho0 = rho1;
...
end
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Structure of the scilab code 4/4Space loop to update the value of the density ρ at each point ofthe grid i
for n = 1:Nt
rho0 = rho1;
for i = 1:Nx
if i == 1 %% Left boundary conditionrho1(i) = ...
elseif i == Nx %% Right boundary conditionrho1(i) = ...
elserho1(i) = ...
end
endend
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
The maximum principleRecall the numerical scheme for the diffusion equation
ρn+1i =
(1− 2
∆t
∆x2
)ρni +
∆t
∆x2ρni+1 +
∆t
∆x2ρni−1 ,
Observation. ρn+1i is a linear combination of ρn
i−1, ρni and ρn
i+1.
In order to guarantee the maximum principle, it is necessary toimpose a condition between ∆t and ∆x :
2∆t
∆x2< 1 .
CFL condition. In this case, ρn+1i is a convex combination of
ρni−1, ρ
ni and ρn
i+1.
Therefore,
∀i = 1 . . .Nx minjρ0j ≤ ρn
i ≤ maxjρ0j
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Conservation of mass
The conservation of mass is automatic when the numerical schemeis written with the flux formulation
ρn+1i − ρn
i
∆t+
F ni+ 1
2
− F ni− 1
2
∆x= 0
To see this, simply sum up the equation:
1
∆t
Nx∑i=1
ρn+1i ∆x − 1
∆t
Nx∑i=1
ρni ∆x + F n
Nx+ 12− F n
12
= 0
The no-flux boundary condition F n12
= F nNx+ 1
2
= 0 guarantees
∑i
ρn+1i ∆x =
∑i
ρni ∆x =
∑i
ρ0i ∆x .
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Contents
1- Diffusion equation
2- The diffusion-transport equation
3- The diffusion-transport equation in self-similar variables
4- Cluster formation
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
The diffusion-transport equation
We switch to the one-dimensional polarization equation.
∂tρ(t, x)− ∂x (µ(t)ρ(t, x)) = ∂xxρ(t, x)
+ Neumann boundary condition at x = 0
+ nonlinear coupling via µ(t) = ρ(t, 0).
Again, we rewrite the equation in ”divergence” form,
∂tρ(t, x) + ∂xF = 0 ,
The flux F is given by: F = −∂xρ− µρ.
The boundary condition at {x = 0} is F (0) = 0 (no-flux boundarycondition).
We assume that the speed µ(t) is nonnegative µ(t) ≥ 0.
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Strategy for computing ρ(t, x) numerically
We replace the equation ∂tρ(t, x) + ∂xF = 0 with the numericaldiscretization
ρn+1i − ρn
i
∆t+
F ni+ 1
2
− F ni− 1
2
∆x= 0
F ni+ 1
2= −
ρni+1 − ρn
i
∆x︸ ︷︷ ︸Fick ′slaw
+ µn ρni+ 1
2︸ ︷︷ ︸transport
.
Important question. How to interpolate the value
ρni+ 1
2= ρ(t, x + ∆x/2) ?
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
The correct choice for ρni+ 1
2
for transport only
Recall the key assumption µn ≥ 0.
Choice 1. ρni+ 1
2
= ρni
ρn+1i =
(1 + µn ∆t
∆x
)ρni − µn ∆t
∆xρni−1 ,
Choice 2. ρni+ 1
2
= ρni+1 [Upwind scheme]
ρn+1i =
(1− µn ∆t
∆x
)ρni + µn ∆t
∆xρni+1 ,
Choice 3. ρni+ 1
2
= 12 (ρn
i + ρni+1) [Centered scheme]
ρn+1i = ρn
i + µn ∆t
2∆xρni+1 − µn ∆t
2∆xρni−1 ,
Only the second choice preserves the maximum principle, under thecondition
µn ∆t
∆x< 1 .
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Combination of the two fluxes
We eventually get
ρn+1i − ρn
i
∆t+
1
∆x
(−ρni+1 − ρn
i
∆x− µnρn
i+1 +ρni − ρn
i−1
∆x+ µnρn
i
)= 0 ,
with suitable boundary conditions for i = 1 and i = Nx .
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Contents
1- Diffusion equation
2- The diffusion-transport equation
3- The diffusion-transport equation in self-similar variables
4- Cluster formation
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Self-similar rescaling
Rewrite the density ρ in self-similar variables,
ρ(t, x) =1√
1 + 2tu
(log√
1 + 2t,x√
1 + 2t
)The equation for u(τ, y) is the following,
∂τu(τ, y)−∂y
(yu(τ, y)+v(τ)u(τ, y)
)= ∂yy u(τ, y) , v(τ) = u(τ, 0) .
We get an additional drift term −∂y (yu(τ, y)) due to the changeof frame (t, x)→ (τ, y).
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Long-time asymptotics
In the sub-critical regime, M < 1, the solution u(τ, y) convergestowards a unique stationary state G , given by
G (y) = α exp
(−αy − y 2
2
).
The parameter α is fixed by the conservation of mass∫y>0
G (y) dy =
∫y>0
u(τ, y) dy =
∫x>0
ρ(t, x) dx = M .
Exercise. Perform the numerical simulations for u(τ, y), solutionto the polarization equation in self-similar variables (the onlydifference is the additional drift term). Compare the long-timebehaviour with the expected value for G .
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Contents
1- Diffusion equation
2- The diffusion-transport equation
3- The diffusion-transport equation in self-similar variables
4- Cluster formation
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
A one-dimensional model for cluster formation
The simplest model for bacterial chemotaxis is the followingcoupled system for the cell density ρ(t, x) and the chemicalconcentration S(t, x).{
∂tρ(t, x) + ∂x (ρ(t, x) u(t, x)) = ∂xxρ(t, x) , x ∈ R
−∂xxS(t, x) = ρ(t, x)
u(t, x) = −∫
v∈Vvφ (v∂xS(t, x)) dv
For simplicity we set V = (−1, 1).
The function φ depends on individual features of the bacteria (e.g.the way they react to the chemical signal).
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
Cluster formation
Exercise. Perform the numerical simulations for the couple (ρ, S)for different choices of function φ:
• φ(Y ) = −Y , i.e. the Keller-Segel model,
• φ(Y ) = −sign (Y ), i.e. a step function: it is a good model forchemotaxis in bacteria populations.
1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation
References
V. Calvez, N. Meunier and R. Voituriez, C. R. Math. Acad. Sci. Paris(2010)
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan, B.
Perthame, PLoS Comput Biol (2010)
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