Problem and objectives
Hyperbolic model
Full model
Numerical results
A new Well-Balanced scheme for a hyperbolic
model of chemotaxis
Christophe Berthon, Anaïs Crestetto and Françoise Foucher
LMJL, Université de Nantes
ANR GEONUM
Deuxième école EGRIN
July 1, 2014
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
1/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Hyperbolic model of chemotaxis
∂t ρ + ∂x (ρu) = 0,
∂t (ρu) + ∂x ρu 2 + p = χρ∂x φ − αρu,
∂t φ − D∂xx φ = aρ − bφ,
(1)
(2)
(3)
where
- ρ (x, t) ≥ 0: particles density,
- u (x, t) ∈ R: mean velocity,
- φ (x, t) ≥ 0: concentration of chemoattractant,
- p (ρ) = εργ : pressure law, with γ > 1 adiabatic exponent and
ε > 0 a constant,
- χ ≥ 0, α ≥ 0, D > 0, a > 0 and b > 0 some parameters.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
2/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Objectives
• Preserve equilibrium states (at rest, with u = 0), that are
given by
ε γ γ−1
− φ = K,
χ γ−1 ρ
(4)
−D∂xx φ = aρ − bφ,
with a constant K .
• W-B scheme on the hyperbolic part (1)-(2)1,2 , and also on the
equation (3) for φ.
1
2
Natalini, Ribot, Twarogowska, CMS 2014.
Twarogowska, PhD Thesis 2011.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
3/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Outline
• Hyperbolic model
• Full model
• Numerical results
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
4/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
Hyperbolic model
We first look at the two first equation (1)-(2), that we rewrite as
∂t w + ∂x F (w ) = S (w )
(5)
with
ρ
ρu
0
w=
, F (w ) =
and S (w ) =
,
ρu
ρu 2 + p
χρ∂x φ − αρu
considering φ as a known source term.
• First study: without friction (α = 0).
• Second study: with friction (α > 0).
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
5/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
Approximate Riemann solver
Approximate Riemann solver w̃ ( xt , wL , wR ) defined by:
• velocities λL < 0 < λR :
−
+ + +
λL = min(0− , λ−
L , λR ) and λR = max(0 , λL , λR ),
±
where λ±
L and λR denote the eigenvalues of the flux Jacobian
matrix:
p
p
λ± = u ± c where c = c(ρ) = P ′ (ρ) = εγργ−1 ,
• intermediate states wL⋆ and wR⋆ (will be defined later),
• the CFL condition:
∆t
∆x
max (|λL |, |λR |) ≤ 12 .
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
6/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
HLL consistency condition
Consistency condition from Harten, Lax and van Leer:
Z ∆x
Z ∆x
2
2
1
x
x
1
w̃ ( , wL , wR )dx =
wR ( , wL , wR )dx,
∆x − ∆x
∆t
∆x − ∆x
∆t
2
2
{z
} |
{z
}
|
Ã
x
wR ( t , wL , wR )
AR
with
the exact solution of the Riemann problem.
• Left side:
∆t
1
(λL (wL − wL⋆ ) + λR (wR⋆ − wR )) .
à = (wL + wR ) +
2
∆x
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
7/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
∂t w + ∂x F (w ) = S (w )
(5)
∆x
the exact
• Right side (by integrating on [0, ∆t] × − ∆x
2 , 2
equation (5)) in the case α = 0:
1
∆t
0
AR = (wL + wR ) −
(F (wR ) − F (wL )) + ∆t
.
SR
2
∆x
R ∆t R ∆x
1
2
χρ∂x φdxdt
• Exact source term SR = ∆t∆x
0
− ∆x
approximated by S ⋆ .
• Four other unknowns: wL⋆ =
→ We need 5 equations.
C. Berthon, A. Crestetto, F. Foucher
2
ρ⋆L
ρ⋆L uL⋆
and wR⋆ =
ρ⋆R
⋆
ρR uR⋆
W-B scheme for a model of chemotaxis
.
8/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
• Two first equation: Ã = AR ⇔
λL (ρL − ρ⋆L ) + λR (ρ⋆R − ρR ) = ρL uL − ρR uR ,
λL (ρL uL − ρ⋆L uL⋆ ) + λR (ρ⋆R uR⋆ − ρR uR ) = ρL uL2 + pL − ρR uR2 − pR + ∆xS ⋆ .
• Third equation: ρ⋆L uL⋆ = ρ⋆R uR⋆ =: q ⋆ .
• Two last equations: study the steady states at rest.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
9/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
Steady states at rest
• Steady states at rest associated to (5) given by:
where e (ρ) is defined by ∂x e =
e(ρ) = ε
1
ρ
u = 0,
e − χφ = K ,
∂x P which is, since P(ρ) = εργ :
γ
ργ−1 + e0
γ−1
with e0 an arbitrary constant.
uL = uR = 0,
• In the Riemann problem:
eL − χφL = eR − χφR = K .
• Approximated Riemann solver preserves steady states at rest:
∗
uL = uR∗ = 0 (⇒ q ∗ = 0),
eL∗ − χφL = eR∗ − χφR = K .
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
10/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
Source-term approximation3
• We are at steady state ⇒ Ã = AR becomes
(
λL (ρL − ρ⋆L ) + λR (ρ⋆R − ρR ) = 0,
(λR − λL )q ⋆ + pR − pL = ∆x S ⋆ .
• We want to preserve steady states ⇒ we have to ensure q ⋆ = 0
⇒ we suggest to put the following consistent expression of S ⋆ :
S⋆ =
χ pR − pL
(φR − φL ) .
∆x eR − eL
→ Fourth equation.
→ Necessary condition for Well-Balanced property.
3
Berthon, Chalons, submitted.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
11/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
• For the last equation, we choose to impose
eL
ρ⋆
ρ⋆L
− χφL = eR R − χφR ,
ρL
ρR
which is consistent with eR − eL = φR − φL and gives a
linearization of the equation ∂x e = χ∂x φ.
→ Sufficient condition for Well-Balanced property.
• Solving the system of five equations ⇒ expressions for ρ⋆L , ρ⋆R ,
q ⋆ and S ⋆ .
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
12/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
Case with friction: α > 0
• What changes? The second equation:
∂t (ρu) + ∂x ρu 2 + p = χρ∂x φ − αρu.
∆x
• Integrating on [0, ∆t] × − ∆x
gives
2 , 2
Z ∆t
∆t
1
2
2
ρR uR +pR −ρL uL −pL +∆tSR −α F (t) dt,
F (∆t) = (ρL uL +ρR uR )−
2
∆x
0
where
1
F (t) =
∆x
Z
∆x/2
−∆x/2
(ρu)R (x, t) dx.
→ Equation in F (∆t):
F ′ (∆t) =
∆t
1
(. . . ) −
(. . . ) + ∆tSR − αF (∆t) .
2
∆x
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
13/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
• Solution:
F (∆t) = Ke
with
−α∆t
1
1
2
2
+
ρR uR + pR − ρL uL − pL + SR ,
−
α
∆x
1
1
1
2
2
K = (ρL uL + ρR uR ) −
ρR uR + pR − ρL uL − pL + SR .
−
2
α
∆x
• HLL consistency condition: Ã2 = F (∆t).
• Approximation of SR in order to preserve equilibrium (u = 0):
S ⋆ defined previously suits.
• ρ⋆L and ρ⋆R not changed.
• q ⋆ is the only quantity that has to be modified when taking
α > 0.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
14/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver without friction
Construction of the approx. Riemann solver with friction
Correction for the positivity
Correction for the positivity
Min-Max procedure in order to ensure ρ⋆L ≥ 0 and ρ⋆R ≥ 0, where
λR R
ρL uL − ρR uR
− δR
,
δR λL − δL λR
δR λL − δL λR
ρL uL − ρR uR
λL R
− δL
,
ρ⋆R = ρR +
δR λL − δL λR
δR λL − δL λR
ρ⋆L = ρL +
with δR :=
eR
ρR ,
δL :=
eL
ρL
and R := χ (φR − φL ) − eR .
• For simplicity reasons: λR = −λL .
• Consistency gives now:
Ã1 = AR,1 ⇔ ρ⋆L + ρ⋆R = ρL + ρR −
• We take:
ρR uR − ρL uL
=: 2ρHLL .
λR
ρ⋆R = min (max (0, ρ⋆R ) , 2ρHLL ) ,
ρ⋆L = min (max (0, ρ⋆L ) , 2ρHLL ) .
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
15/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver
Full numerical scheme
W-B property
Full model
We now look at the equation (3) on φ, that we rewrite as:
∂t φ − D∂x ψ = aρ − bφ,
where ρ is assumed to be known (since computed previously) and
ψ := ∂x φ.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
16/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver
Full numerical scheme
W-B property
HLL consistency condition
Consistency condition from Harten, Lax and van Leer:
1
∆x
|
Z
Z ∆x/2
x
x
1
, φL , φR dx =
, φL , φR dx ,
φ̃
φR
∆t
∆x −∆x/2
∆t
−∆x/2
{z
} |
{z
}
∆x/2
Ã
AR
with φR the exact Riemann solver and φ̃ the approximated one.
• Left side:
à =
1
∆t
(φL + φR ) +
(λL (φL − φ⋆L ) + λR (φ⋆R − φR )) .
2
∆x
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
17/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver
Full numerical scheme
W-B property
∂t φ − D∂x ψ = aρ − bφ
(3)
∆x
• Right side (by integrating on [0, ∆t] × − ∆x
the exact
2 , 2
equation (3)):
Z ∆t Z ∆x/2
1
D∆t
1
AR = (φL + φR ) +
(ψR − ψL ) +
aρR − bφR dxdt.
2
∆x
∆x 0
−∆x/2
• First part of the model:
Z ∆x/2
t
1
1
(ρR uR − ρL uL ) .
ρR (x, t) dx = (ρL + ρR ) −
∆x −∆x/2
2
∆x
• We get:
1
D∆t
(φL + φR ) +
(ψR − ψL )
2
∆x
Z ∆t
∆t
∆t 2
F (t) dt.
(ρL + ρR ) −
(ρR uR − ρL uL ) − b
+a
2
2∆x
0
F (∆t) := AR =
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
18/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver
Full numerical scheme
W-B property
→ Equation on F (∆t):
1
∆t
D
′
(ψR − ψL )+a
(ρL + ρR ) −
(ρR uR − ρL uL ) .
F (∆t)+bF (∆t) =
∆x
2
∆x
• Solution:
1
(φL + φR ) e −b∆t + α∆t + β 1 − e −b∆t
2
a
with α = − b∆x (ρR uR − ρL uL ) and
F (∆t) =
β=
D
b∆x
a
(ψR − ψL )+ 2b
(ρL + ρR )+ b2a∆x (ρR uR − ρL uL ).
• Consistency condition: Ã = AR = F (∆t).
+ Choice: φL − φ⋆L = φR − φ⋆R .
→ Expressions for φ⋆L and φ⋆R .
• And what about ψL and ψR ? Answer later...
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
19/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver
Full numerical scheme
W-B property
Finite Volumes scheme
• 1D domain [0, L] discretized by N + 1 points: xi = i∆x,
i = 0, . . . , N, ∆x = NL .
R xi +1/2
1
n
• Evolution in time of win ≈ ∆x
xi −1/2 w (x, t ) dx and
R
xi +1/2
1
n
φni ≈ ∆x
xi −1/2 φ (x, t ) dx, where
t n = n∆t for a time step ∆t.
• Scheme coming from the previously defined Riemann solvers:

n+1
∆t
 wi
= win − ∆x
λi − 1 ,R win − wi⋆− 1 ,R − λi + 1 ,L win − wi⋆+ 1 ,L ,
2 2 2 2
n
⋆
n
⋆
 φn+1 = φn − ∆t λ 1
.
i
i
i − ,R φi − φi − 1 ,R − λi + 1 ,L φi − φi + 1 ,L
∆x
2
C. Berthon, A. Crestetto, F. Foucher
2
2
W-B scheme for a model of chemotaxis
2
20/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver
Full numerical scheme
W-B property
Definition of ψin
• ψin such that good approximation of (∂x φ)ni , for example of
the form
1
φni+1 − φni−1 × 1(∆x)
ψin =
2∆x
where 1(∆x) has to be consistent with 1 when ∆x tends to 0.
= φni at equilibrium in
• 1(∆x): correction term such that φn+1
i
the particular case γ = 2.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
21/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver
Full numerical scheme
W-B property
Equilibrium for γ = 2
• Equilibrium given by:

(
 u = 0,
u = 0,
ρ + K,
φ = 2ε
⇒
χ
∂xx ρ −

D∂xx φ − bφ = −aρ,
if ρ = 0 :
χ
2εD
2εb
χ
χ
− a ρ = Kb 2εD
.
• Solutions1,2 :
if ρ > 0, C < 0 :
if ρ > 0, C > 0 :
√ √ φ (x) = A cosh x b + B sinh x b ,
p p φ (x) = A cos x |C | + B sin x |C | − φp ,
√ √ φ (x) = A cosh x C + B sinh x C − φp ,
where A and B are some constants, C =
1
2
1
D
Natalini, Ribot, Twarogowska, CMS 2014.
Twarogowska, PhD Thesis 2011.
C. Berthon, A. Crestetto, F. Foucher
b−
aχ 2ε
ρ (x) =
χ
2ε
ρ (x) =
χ
2ε
and φp =
(φ (x) − K ) ,
(φ (x) − K ) ,
Kaχ
2εb−aχ .
W-B scheme for a model of chemotaxis
22/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Construction of the approx. Riemann solver
Full numerical scheme
W-B property
• Injecting these solutions in the Finite Volumes scheme and
= φni gives the appropriate expression of 1(∆x):
imposing φn+1
i
if ρ = 0 :
1(∆x) =
if ρ > 0, C < 0 : 1(∆x) =
if ρ > 0, C > 0 : 1(∆x) =
∆x 2
√b
2 cos( b∆x )−1 ,
∆x 2
√ C
,
2 cos
|C |∆x −1
∆x 2
√C
2 cosh( C ∆x )−1 .
• Expression of ψin
ψin =
1
φni+1 − φni−1 × 1(∆x)
2∆x
with this 1(∆x) ⇒ steady states exactly preserved for γ = 2
and approximated for γ > 2.
• Min-Max procedure in order to ensure φ ≥ 0.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
23/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Testcase 1
Testcase 2
Testcase 1: perturbation of an equilibrium solution
• Exact equilibrium solution1,2 :
√

2εbK cos(√τ x )

− τaK
for x ∈ [0, x ] ,

D,

x)
 τ χD cos( τq
b
φ (x) =
(x−L)
cosh
D

2εK
, for x ∈ ]x, L] ,

q
−

χ

b
(x−L)
D
cosh
ρ (x) =
(
χ
2ε φ (x)
0,
+
D
Mτ 3/2 √
√
b tan( τ x )− τ x ,
u (x) = 0,
where τ =
1
2
1
D
aχ
2ε
− b and x s.t.
q
b
τD
for x ∈ [0, x] ,
for x ∈ ]x, L] ,
q
√
b
tan ( τ x) = tanh
D (x − L) .
Natalini, Ribot, Twarogowska, CMS 2014.
Twarogowska, PhD Thesis 2011.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
24/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Testcase 1
Testcase 2
ρ and φ with a = b = D = ε = 1, γ = 2, χ = 50, L = 1, ∆x = 0.01.
6
6
Steady state
5
5
4
4
ρ
ρ
T=0
3
3
2
2
1
1
0
0
0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
x
0.6
1
1.6
T=0
Steady state
1.4
1.4
1.2
1.2
1
1
0.8
0.8
φ
φ
0.8
x
1.6
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
x
C. Berthon, A. Crestetto, F. Foucher
0
0.2
0.4
0.6
0.8
1
x
W-B scheme for a model of chemotaxis
25/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Testcase 1
Testcase 2
Testcase 2: influence of parameters on the steady state
• Initial conditions: ρ (x, 0) = 1 + sin 4π|x − L4 | ,
u (x, 0) = 0,
φ (x, 0) = 0.
• Study
• Density ρ as a function of x at steady state.
• Influence of L and χ with a = b = D = ε = 1.
• Influence of γ with a = 20, b = 10, D = 0.1, ε = 1.
• Validation? No analytical solution. But results similar to those
of Twarogowska1 ,2 .
1
2
Natalini, Ribot, Twarogowska, CMS 2014.
Twarogowska, PhD Thesis 2011.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
26/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Testcase 1
Testcase 2
Influence of L at γ = 2, χ = 3, ∆x = 0.01.
0.7
L=1, χ=3
1.4
L=5, χ=3
0.6
1.2
0.5
1
0.4
ρ
ρ
0.8
0.3
0.6
0.2
0.4
0.1
0.2
0
0
0
0.2
0.4
0.6
0.8
0
1
1
2
x
3
4
5
x
0.7
0.3
L=7, χ=3
0.6
L=30, χ=3
0.25
0.5
0.2
ρ
ρ
0.4
0.15
0.3
0.1
0.2
0.05
0.1
0
0
0
1
2
3
4
5
6
7
x
C. Berthon, A. Crestetto, F. Foucher
0
5
10
15
20
25
30
x
W-B scheme for a model of chemotaxis
27/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Testcase 1
Testcase 2
Influence of χ at γ = 2, L = 7, ∆x = 0.01.
0.7
0.8
L=7, χ=3
L=7, χ=5
0.7
0.6
0.6
0.5
0.5
ρ
ρ
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
1
2
3
4
5
6
0
7
1
2
3
x
4
5
6
7
x
3.5
8
L=7, χ=50
L=7, χ=200
7
3
6
2.5
5
ρ
ρ
2
4
1.5
3
1
2
0.5
1
0
0
0
1
2
3
4
5
6
7
x
C. Berthon, A. Crestetto, F. Foucher
0
1
2
3
4
5
6
7
x
W-B scheme for a model of chemotaxis
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Problem and objectives
Hyperbolic model
Full model
Numerical results
Testcase 1
Testcase 2
Influence of γ at χ = 10, L = 3, ∆x = 0.01.
30
8
γ=2
γ=3
7
25
6
20
ρ
ρ
5
15
4
3
10
2
5
1
0
0
0
0.5
1
1.5
2
2.5
0
3
0.5
1
x
1.5
2
2.5
3
x
3.5
1.8
γ=4
γ=5
1.6
3
1.4
2.5
1.2
2
ρ
ρ
1
0.8
1.5
0.6
1
0.4
0.5
0.2
0
0
0
0.5
1
1.5
2
2.5
3
x
C. Berthon, A. Crestetto, F. Foucher
0
0.5
1
1.5
2
2.5
3
x
W-B scheme for a model of chemotaxis
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Problem and objectives
Hyperbolic model
Full model
Numerical results
Testcase 1
Testcase 2
Conclusions...
• HLL consistent Riemann solver.
• Positivity of ρ and φ.
• Equilibrium states exactly preserved when γ = 2 and well
approached when γ > 2.
• No problem with vacuum ρ = 0.
• We can prove that the scheme is AP in the case α > 0.
... and perspectives
• Understand the behaviour of the asymptotic-parabolic model.
• Extension of the model to the 2D.
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
30/31
Problem and objectives
Hyperbolic model
Full model
Numerical results
Testcase 1
Testcase 2
References
- C. Berthon, C. Chalons, A Fully Well-Balanced, Positive and Entropy-satisfying
Godunov-type method for the shallow-water equations, submitted, HAL:
hal-00956799.
- R. Natalini, M. Ribot and M. Twarogowska, A Well-Balanced Numerical Scheme for
a One Dimensional Quasilinear Hyperbolic Model of Chemotaxis, Commun. Math.
Sci. 12 (1), pp. 13-38, 2014.
- M. Twarogowska, Numerical Approximation and Analysis of Mathematical Models
Arising in Cells Movement, PhD Thesis, Università degli studi dell’Aquila, 2011.
Thank you for your attention!
C. Berthon, A. Crestetto, F. Foucher
W-B scheme for a model of chemotaxis
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