OPTPDE Summer School - Challenges in Applied Control and Optimal Design (July 4-8, 2011)

Long-time behavior of scalar convection- diffusion equations:A numerical approach

Alejandro Pozo? (BCAM)pozo@bcamath.org

Introduction

We are interested in the following scalar convection-diffusion equation:{ut −∆u = a · ∇(F (u)), x ∈ RN × (0,∞)

u(x, t) = u0(x), x ∈ RN(1)

where a ∈ RN , F ∈ C(R1) such that F (0) = 0 and u0 ∈ L1(RN ).This equation is a model for the regularizing effect of a viscosity in a hy-

perbolic conservation law. It appears, for example, in the displacement of afluid through a porous media (cf. [2]), when taking capillarity into account;or in contamination between batches in multi-products pipeline transport (cf.[6]).

Our aim is to make a numerical approach to the asymptotic properties ofthis kind of equations. We will consider the case for N = 2, a = (1, 0) andF (u) = |u|q−1u for q ∈ (1,∞).

Existing analytical results

Let us consider the homogeneous non linear case presented above, givenby: {

ut −∆u = ∂x(|u|q−1u), x ∈ R2 × (0,∞)

u(x, 0) = u0(x), x ∈ R2 (2)

For an initial data u0 ∈ L1(R2), we have that

‖u(t)‖p ≤ Cpt−(

1−1p

), ∀t > 0 (3)

for all p ∈ [1,∞] with Cp = Cp(‖u0‖1). We will distinguish three differentsituations:• For 1 < q < 3

2 the asymptotic behavior is given by self-similar entropicsolutions of the following hyperbolic-parabolic equation:{

ut − uyy = ∂x(|u|q−1u), x ∈ R2 × (0,∞)

u(x, 0) = Mδ, x ∈ R2 (4)

where M =∫R2 u0(x)dx. In fact, it holds that

‖u(t)‖∞ ≤ Ct−3/2q. (5)

• For q = 32, the equation (2) admits self-similar solutions of the form

u(x, t) = 1tf(x√t

). In [1, 3, 4], it is shown that f is the unique solu-

tion for

−∆f − x · ∇f2− f = ∂x(|f |1/2f ), x ∈ R2 (6)

such that∫R2 f (x)dx = M and, therefore

t1−1

p‖u(x, t)−MG(x, t)‖p→ 0 when t→ 0 (7)

where G(x, t) = (4πt)−1e−|x|2/4t is the heat kernel.

• For q > 32, the decay rate (7) is sharp and u behaves, as t → ∞, as the

linear Gaussian heat kernel, as shown in [3, 4].

Numerical approach

For the numerical simulation of the three situations, we make use of thecorresponding variational problem. Defining the convective derivative asDtu := ut + v · ∇u, for a fluid velocity v = (q|u|q−1, 0) we have∫

R2[ϕDtu +∇u∇ϕ] dS = 0, ∀ϕ ∈ H1

0(R2) (8)

We use the Characteristics-Galerkin method to approach the convectivederivative. This consists on approaching the convective derivative by an ap-proximation of the characteristics of the fluid velocity:

Dtu(x0) ≈ 1

∆t

(um+1(x0)− um(Xm(x0))

)(9)

where Xm(x0) = χ(t−∆t) ≈ x0−v(x0)∆t and χ is the characteristic curvegiven by χ(t) = v(χ(t), t) with χ(t0) = x0.

Taking all of that into account, we use FreeFem++ [5] to recreate theanalytical behaviors, using different finite elements and initial conditions.

Simulations

Our preliminary results show that the asymptotic behavior also holds forthe numerical approach. Here we show the outcome for two different initialconditions:

(i) u0(x) = 2e−|x/2|2(ii) u0(x) = χ[−2,2]×[−4,4](x)

We use P2 finite elements and restrict the domain to a finite but enoughbig one, [−500, 500]2, whereas t goes from 0 to 100.

For the first example, we can clearly observe the decay (3):

Even if the initial data is discontinuous, the second one also behaves insame way. Besides, we can also appreciate the regularizing effect of the vis-cosity:

Other numerical simulations done still confirm the analytical results de-scribed above.

Future research

On the one hand, there is a lack of theoretical base for the numerical re-sults. We need to assure that what we see on the simulations has a foundationon the discrete version of the problem.

On the other hand, we have to adapt our simulations in order to catchmore analytical results.

References[1] J. Aguirre, M. Escobedo and E. Zuazua. Self-similar solutions of a convection-diffusion equation and related elliptic problems. Communications in Partial Differential Equations, 15(2), (1990), pp. 139-157.

[2] S. E. Buckley and M. C. Leverett. Mechanism of fluid displacements in sands. A. I. M. E., 146, (1942), pp. 107-116.

[3] M. Escobedo and E. Zuazua. Comportement asymptotique des solutions d’une equation de convection-diffusion. Comptes rendus de l’Academie des sciences, 309, (1989), pp.329-334.

[4] M. Escobedo and E. Zuazua. Large time behavior for convection-diffusion equations in RN . Journal of Functional Analysis, 100, (1991), pp. 119-161.

[5] F. Hecht. FreeFem++. Third edition, version 3.12. http://www.freefem.org/ff++/ftp/freefem++doc.pdf

[6] D. Songsheng and P. Jianing. Application of convections-diffusion equation to the analyses of contamination between batches in multi-products pipeline transport. Applied Mathematics and Mechanics, 19(8), (1998),

pp. 757-764.

? This work is supported by the grants PI2010-04 and BFI-2010-339, both by the Basque Government, and by the ERC Advanced Grant FP7-246775 NUMERIWAVES.

I wish to thank Enrique Zuazua and Mehmet Ersoy for their explanations and suggestions.