A finite volume scheme for
nonlinear degenerate parabolic equations
Marianne BESSEMOULIN-CHATARD, Francis FILBET
Laboratoire de Mathématiques - UMR 6620
Université Blaise Pascal, Clermont-Ferrand
Spring School “ Kinetic Theory and Fluid Mechanics ”
Lyon, March 26-30, 2012
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1
Introduction
2
Presentation of the numerical scheme
3
Properties of the scheme
4
Numerical simulations
5
Conclusion
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Introduction
General framework
∂t u = div (f (u)∇V (x) + ∇r(u)) ,
u(x, 0) = u0 (x), x ∈ Ω
x ∈ Ω,
t > 0,
Hypotheses
Ω ⊆ Rd bounded domain or Ω = Rd ,
if Ω ⊂ Rd , Neumann boundary conditions on ∂Ω,
Z
u0 ∈ L1 (Ω), u0 ≥ 0 and
u0 (x) dx = M ,
Ω
r ∈ C 2 (R+ ) such that r(0) = 0 and r′ (u) ≥ 0.
Notations
h(s) :=
Z
1
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s
r′ (τ )
dτ ,
τ
H(s) :=
Z
s
h(τ ) dτ ,
s ≥ 0.
0
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Introduction
Long-time behavior: case f (u) = u
J.A. Carrillo, A. Jüngel, P.A. Markowich, G. Toscani, A. Unterreiter, Entropy
dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities,
Monatsh. Math., 2001.
eq
eq
Stationary equation:
Z u ∇V (x) + ∇r(u ) = 0.
(V u + H(u)) dx.
Entropy: E(u) :=
Ω
ueq
ueq ∈ L1 (Ω) is an equilibrium solution
m
R
is a minimizer of E in C := {u ∈ L1 (Ω), Ω u = M }.
Results
∃! ueq if V is sufficiently regular,
exponential decay of the relative entropy E(t) and its dissipation I(t), where
Z
d
2
u |∇ (V + h(u))| dx.
E(t) := E(u(t)) − E(ueq ), I(t) := − E(u(t)) =
dt
Ω
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Introduction
Examples
The porous media equation
∂t u = div (xu + ∇um ) on Rd × (0, T ).
J.A. Carrillo, G. Toscani, 2000: convergence of the solution to the
Barenblatt-Pattle distribution ueq (x) =
1
m − 1 2 m−1
|x|
C1 −
.
2m
+
Drift-diffusion system for semiconductors

 ∂t N = div (∇r(N ) − N ∇V )
∂t P = div (∇r(P ) + P ∇V ) on Ω × (0, T ).

∆V = N − P − C
Jüngel, 1995: convergence of the solution to the thermal equilibrium
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∆V eq = g(αN + V eq ) − g(αP − V eq ) − C,
N eq = g(αN + V eq ), P eq = g(αP − V eq ).
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Introduction
Long-time behavior: case r(u) = u
Nonlinear Fokker-Planck equation for fermions and bosons
∂t u = div (xu(1 + ku) + ∇u) on Rd × (0, T ),
with k = 1 in the boson case and k = −1 in the fermion case.
1
,
Stationary solution: ueq (x) =
|x|2
2
−
k
βe
Z 2
|x|
Entropy: E(u) :=
u + u log(u) − k(1 + ku) log(1 + ku) dx,
2
Rd
2
2
Z
u
|x|
dx.
+ log
Dissipation: I(t) =
u(1 + ku) ∇
2
1 + ku
Rd
Exponential decay towards the equilibrium solution:
in any dimension for fermions (J.A. Carrillo, P. Laurençot, J. Rosado, 2009),
in 1D for bosons (J.A. Carrillo, J. Rosado, F. Salvarani, 2008).
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Introduction
Long-time behavior: discretizations
Temporal semi-discretizations: proof of exponential decay of E(t).
◮
◮
◮
A. Arnold, A. Unterreiter, 2001 (linear Fokker-Planck),
J.A. Carrillo, M. Di Francesco, M.P. Gualdani, 2007 (nonlinear diffusion),
M. Burger, J.A. Carrillo, M.T. Wolfram, 2010 (nonlinear diffusion).
Full discretizations: proof of convergence to the equilibrium (drift-diffusion
system for semiconductors).
◮
◮
C. Chainais-Hillairet, F. Filbet, 2007 (nonlinear upwind scheme),
M. B.-C., 2011 (Scharfetter-Gummel scheme).
0
10
0
Upwind
Upwind nonlin
SG
10
−5
10
−10
I(t)
E(t)
10
−10
10
−20
10
−15
Upwind
10
Upwind nonlin
SG
−20
10
−30
0
1
2
3
4
t
(a) Relative energy E(t)
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5
10
0
1
2
3
4
5
t
(b) Dissipation I(t)
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Introduction
Aim
Construct a finite volume scheme for nonlinear parabolic equations which:
is second order accurate in space, even in the degenerate case,
preserves steady-states,
provides a discrete entropy estimate.
Main ideas:
Discretize together convective and diffusive parts (long-time behavior).
Use slope-limiters (second-order accuracy).
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Presentation of the numerical scheme
Presentation of the numerical scheme
Finite volume scheme
∆xi
•
xi− 1
2
•
Ki
•
•
xi+ 1
xi
•
xi+ 3
xi+1
2
2
∆xi+ 1
2
∆t: time step.
Finite volume scheme
∆xi
Uin+1 − Uin
n
n
+ Fi+
1 − F
i− 21 = 0,
2
∆t
where
Fi+ 12 ≈ −(f (u)∂x V + ∂x r(u))|xi+ 1 .
2
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Presentation of the numerical scheme
Case of a linear convection (f (u) = u)
− (u∂x V + ∂x r(u)) = −∂x (V + h(u)) u.
{z
}
|
“velocity”
−∂x (V + h(u)) |xi+ 1 ≈ Ai+ 21 = −dVi+ 12 − dh(U )i+ 12 ,
2
with
dVi+ 21 =
V (xi+1 ) − V (xi )
,
∆xi+ 21
dh(U )i+ 12 =
h(Ui+1 ) − h(Ui )
∆xi+ 21
Classical upwind flux ⇒ Fi+ 12 = A+
U − A−
U
i+ 1 i
i+ 1 i+1
2
2
where x+ = max(0, x), x− = max(0, −x).
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Presentation of the numerical scheme
Case of a linear convection (f (u) = u)
Second-order accuracy: Ui ← Ui+ 21 ,− ,
where θi =
φ=0
φ(θ) =
Ui+ 12 ,−
=
Ui+ 12 ,+
=
Ui+1 ← Ui+ 21 ,+ , with
1
Ui + φ(θi )(Ui+1 − Ui ),
2
1
Ui+1 − φ(θi+1 )(Ui+2 − Ui+1 ),
2
Ui − Ui−1
, and φ slope limiter:
Ui+1 − Ui
Upwind, order 1,
θ + |θ|
1 + |θ|
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Van Leer, order 2.
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Presentation of the numerical scheme
General case
− (f (u)∂x V + ∂x r(u)) = −∂x V + h̃(u) f (u),
where h̃(u) is such that h̃′ (u)f (u) = r′ (u).
Local Lax-Friedrichs flux ⇒
Fi+ 12 =
Ãi+ 12
2
(f (Ui ) + f (Ui+1 )) −
Ãi+ 21 αi+ 12
2
(Ui+1 − Ui ),
with
Ãi+ 12 = −dVi+ 21 − dh̃(U )i+ 12 ,
and
αi+ 12 = max (|f ′ (u)|) , u between Ui and Ui+1 .
Order 2 : Ui ← Ui+ 21 ,− ,
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Ui+1 ← Ui+ 12 ,+ .
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Properties of the scheme
Properties of the scheme
The semi-discrete scheme
1 d
Fi+ 21 − Fi− 21 = 0.
Ui +
dt
∆xi
We assume that f (u) = u.
Results:
The scheme preserves the nonnegativity of Ui (t).
The scheme preserves the equilibrium
dh(U )i+ 12 + dVi+ 12 = 0 =⇒ Fi+ 21 = 0.
We have the following entropy estimate: for all 0 < t1 ≤ t2 < ∞,
Z t2
I∆ (t) dt ≤ E∆ (t1 ),
0 ≤ E∆ (t2 ) +
t1
where E∆ (t) and I∆ (t) are semi-discrete versions of the relative entropy and
its dissipation.
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Properties of the scheme
The fully-discrete scheme
m(Ki )
Uin+1 − Uin
n
n
+ Fi+
1 − F
i− 21 = 0.
2
∆t
We assume that f (u) = u.
The scheme preserves the equilibrium.
For n ≥ 0, we assume that Uin ≥ 0 for all i.
Then under the CFL condition
1
n
) − h(Uin ) ≤ min ∆x2i ,
∆t max V (xi+1 ) − V (xi ) − h(Ui+1
i
2 i
the scheme preserves the nonnegativity:
Uin+1 ≥ 0 pour tout i.
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Numerical simulations
Numerical simulations
Other numerical fluxes
Fi+ 12 ≈ −(u∂x V + ∂x r(u))|xi+ 1
2
Classical upwind (CU) Eymard, Gallouët, Herbin, 2000,
+
−
r(Ui+1 ) − r(Ui )
Ui − −dVi+ 21
Ui+1 −
Fi+ 12 = −dVi+ 12
.
∆xi+ 12
Scharfetter-Gummel extended (SG) M. B.-C., 2011,
Fi+ 12 =
dri+ 21
∆xi+ 21
where dri+ 12 =
and B(x) =
B
∆xi+ 12 dVi+ 21
dri+ 12
!
Ui − B
−∆xi+ 12 dVi+ 21
dri+ 12
!
Ui+1
!
,
h(Ui+1 ) − h(Ui )
,
log(Ui+1 ) − log(Ui )
x
, x 6= 0, B(0) = 1 is the Bernoulli function.
ex − 1
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Numerical simulations
Order of convergence
Ω = (−1, 1), T = 0.1, ∂x V = 1, f (s) = s, r(s) = s2 .
∆t = 10−8 , ∆x = 0.04 × 2−j , j = 0, ..., 5.
−4
−6
log(||u−uδ||1)
−8
−10
CU
−12
SG
FU1
FU2
−14
log(∆x)
log(∆x2)
−16
−7
−6.5
−6
−5.5
−5
−4.5
−4
−3.5
−3
log(∆x)
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Numerical simulations
Order of convergence
Ω = (−1, 1), T = 0.01, ∂x V = 1, f (s) = s, r(s) =
∆t = 10−8 , ∆x = 0.04 × 2−j , j = 0, ..., 5.
(s − 1)3
0
if s ≥ 1,
elsewhere.
−7
−8
−9
log(||u−uδ||1)
−10
−11
−12
CU
SG
−13
FU1
FU2
−14
log(∆x)
−15
−16
−7
log(∆x1.9)
−6.5
−6
−5.5
−5
−4.5
−4
−3.5
−3
log(∆x)
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Numerical simulations
Drift-diffusion for semiconductors

 ∂t N
∂t P

∆V
=
=
=
div (∇r(N ) − N ∇V )
div (∇r(P ) + P ∇V )
N −P −C
Ω = (0, 1), T = 2, r(s) = s2 , ∆t = 5.10−5, uniform mesh of 64 cells,
C = −1 if x ≤ 0.5, +1 if x > 0.5.
1
0.8
0.6
0.4
0.2
0
0
Neq
N
0.2
0.4
0.6
0.8
1
x
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Numerical simulations
Drift-diffusion for semiconductors
Ω = (0, 1), uniform mesh of 64 cells,
∆t = 5.10−5 , T = 10.
Relative energy
Energy dissipation
10
0
−5
I(t)
E(t)
10
−10
10
10
−10
CU
−15
10
CU
SGext
SGext
10
FU1
−20
FU2
0
2
FU1
FU2
4
6
t
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8
10
0
2
4
6
8
10
t
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Numerical simulations
The porous media equation
∂t u = div(xu + ∇r(u)), with r(s) = s4 ,
Ω = (−10, 10) × (−10, 10), 50 × 50 Cartesian grid,
∆t = 10−4 , T = 4.
Barenblatt-Pattle ueq .
Evolution of u.
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
10
0
10
5
10
5
0
−5
0
−5
5
10
5
0
0
−5
−5
−10
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−10
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Numerical simulations
The porous media equation
Ω = (−10, 10) × (−10, 10), 200 × 200 Cartesian grid,
∆t = 10−4 , T = 10.
Relative entropy
Entropy dissipation
0
10
10
10
−5
I(t)
E(t)
−5
10
0
−10
10
CU
10
SGext
−10
SGext
FU1
FU1
FU2
FU2
−15
10
0
10
2
CU
4
6
t
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8
10
−15
0
2
4
6
8
10
t
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Numerical simulations
Equation for fermions
∂t u = div(xu(1 − u) + ∇u),
Ω = (−8, 8)3 , 40 × 40 × 40 Cartesian grid,
∆t = 10−4 , T = 3.
Evolution of u(t, x, y, z) = 0.1.
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Equilibrium ueq (x, y, z) = 0.1.
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Numerical simulations
Equation for fermions
Ω = (−8, 8)3 , 40 × 40 × 40 Cartesian grid,
∆t = 10−4 , T = 10.
10
10
0
−5
E(t)
10
I(t)
||u(t)−ueq||1
−10
0
2
4
6
8
10
t
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Numerical simulations
Buckley-Leverett equation
∂t u + ∂x f (u) = ε∂x (ν(u)∂x u) ,
Ω = (0, 1), uniform mesh of 100 cells,
∆t = 10−4 , T = 0.5,
2
f (u) = u /(u2 + (1 − u)2 ), ν(u) = 4u(1 − u).
1
1
0.8
0.8
0.6
t=0
0.4
u(x)
u(x)
0.6
t=0.2
t=0.1
t=0.5
0.2
0
0
t=0
0.4
t=0.2
t=0.1
t=0.5
0.2
0.2
0.4
0.6
x
(c) ε = 10−1
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0.8
1
0
0
0.2
0.4
0.6
0.8
1
x
(d) ε = 10−2
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Numerical simulations
Buckley-Leverett equation
∂t u + ∂x f (u) = ε∂x (ν(u)∂x u) ,
Ω = (0, 1), uniform mesh of 100 cells,
∆t = 10−4 , T = 0.5,
2
f (u) = u /(u2 + (1 − u)2 ), ν(u) = 4u(1 − u).
0.8
0.8
0.6
0.6
t=0
0.4
u(x)
1
u(x)
1
t=0.1
t=0.2
t=0.5
0.2
0
0
t=0
0.4
t=0.2
t=0.1
t=0.5
0.2
0.2
0.4
0.6
x
(e) ε = 10−3
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0.8
1
0
0
0.2
0.4
0.6
0.8
1
x
(f) ε = 0
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Conclusion
Conclusion
Construction of a finite volume scheme for nonlinear parabolic equations which:
preserves steady-states,
preserves the relative entropy decay to zero,
is valid and second-order accurate even in the degenerate case.
Work in progress:
In the case of linear convection, prove the exponential decay of the relative
entropy.
Prove a discrete entropy estimate for the nonlinear Fokker-Planck equation
for bosons and fermions.
Consider the 3D boson case (G. Toscani, 2011).
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