A numerical schema for the transport of nutrients and hormones in plant growth

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DOI 10.1007/s13370-012-0129-z

A numerical schema for the transport of nutrients and hormones in plant growth

S. Boujena · A. Chiboub · J. Pousin

Received: 24 April 2012 / Accepted: 30 November 2012

Abstract Classical numerical methods exhibit numerical discrepancies, when we are dealing with the transport equation in domain of heterogeneous sizes. In this work, a numerical scheme, based on a domain decomposition strategy is built to avoid numerical discrepancies.

Let us mention that this work is inspired from the results given in Picq and Pousin (Variational reduction for the transport equation and plants growth, 2007).

Keywords Plant growth · Branch · Two-dimensional domain · Finite differences · Asymptotic development · Variational reduction · Algorithms · Numerical simulations

1 Introduction

The plant growth presents a seasonal periodicity. The observation of the growth of the principal branch and formation of new buds show that the distance between them is approx- imately the same. Dynamical models of this phenomena have been studied in [4] where the authors had represented a growing plant as a system of intervals, which are called branches.

The branches appear and grow according to some rules. The growing part of the plant contains a narrow exterior part, considered as a surface, where cells proliferate. The displacement of this surface corresponds to the plant growth. The plant growth is due to the nutrients produced in the roots and transported through the whole of the plant and the appearance of new branches is consequence of the concentration of many hormones. The most important of them are cytokinin and Auxin. The Auxin is produced in the growing parts of the plant. And the

S. Boujena (

B

⁾^·^{A. Chiboub}

Sciences Faculty, University Hassan II Casablanca, POB 5366, Maarif, 20200 Casablanca, Morocco e-mail: boujena@yahoo.fr; s.boujena@fsac.ac.ma

A. Chiboub

e-mail: chiboubatika@yahoo.fr J. Pousin

Institut Camille Jordan, CNRS UMR 5208, Université Lyon I, 69200 Villeurbanne Cedex, France e-mail: jerome.pousin@insa-lyon.fr

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Fig. 1 A representation of plant as a heterogeneous domain Q

Trunk

Cytokinin is produced in the roots and in the growing parts of the plant. The concentrations of the nutrients and hormones are modelized by convection-diffusion equations. In [2] and [3], the one dimensional case without diffusion has been studied, and a numerical scheme has been proposed. In this work, the two dimensional case is considered, that is to say, the transport equation is handled in a domain of heterogeneous size. Using only a strategy of domain decomposition, the simulation of the nutrients transport by finite differences method exhibits numerical discrepancies in the thin part of the domain representing the branche.

Thanks to the asymptotic partial decomposition method, we use a perturbation argument in order build an adapted basis to avoid these numerical discrepancies and the transport equation can be reduced to a partial differential equation with less variables in this part of the domain. The asymptotic partial decomposition method introduced in [6] is used in this work for the transport equation in an heterogeneous domain with a general right hand. A numerical method based on finite elements for such models is given in [5]. We show in this work, that the finite differences method works better on the model obtained by the partial asymptotic decomposition strategy than that obtained by the domain decomposition only. The article is organized as follows. The introduction is ended by recalling the mathematical model. In Sect. 2, the asymptotic domain decomposition formulation of the problem is analyzed. In Sect. 3, some numerical results are presented.

Let us introduce the transport equation in the two-dimensional domain Q which represents a part of the plant (Fig. 1).

Let us give some useful notations.

Q = (0, 1) × S

1

∪ 1

2 − , 1 2 +

× S

2

= Q

1

∪ Q

2

, where S

1

= (0, 1)×(0, γ ) et S

2

=

₁

2

− η,

¹₂

+ η

×(γ, 1). Let f ∈ C

⁰

( Q; R), a(t, x, y) = a

^x

(t , x , y )

a

^y

( t , x , y )

and β =

⎛

⎝ 1

a

^x

( t , x , y ) a

^y

(t, x, y)

⎞

⎠ be such that a

^x

∈ C

¹

(Q; R

⁺

) and a

^y

∈ C

¹

(Q; R

⁺

).

In the following, we assume a

^x

and a

^y

to be bounded from below by two positive numbers, and denote by (·/·) the Euclidean inner product. Let ∂ Q

₋

= {(t, x , y), (β/n) < 0}. We look for

u ∈ H (β, Q) = {ρ ∈ L

²

(Q), (β/∇ρ) ∈ L

²

(Q), ρ\∂ Q

₋

∈ L

²

(∂ Q

₋

, |(β/n)|dσ )}

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verifying

(β/∇ u ) =

^∂u_∂t

+ a

^x

( t , x , y )

^∂u_∂x

+ a

^y

( t , x , y )

^∂u_∂y

= f ( t , x , y ) in Q

u = 0 on ∂Q

−

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Theorem 1 The problem (1) has an unique solution u . Proof For a proof the reader is referred to [1].

2 The decomposed problem

In this section, the problem (1) is formulated in the context of the partial asymptotic decomposition method. First, let us build the vector space of finite dimension well suited for the decomposition of the problem (1). Here, variable x ∈ (

¹₂

− η,

¹₂

+ η) is considered as a parameter. For a positive regular function ν, we will denote by L

²

(·, ν dt) the Hilbert’s space equipped with the inner product with the weight function ν.

Lemma 1 Suppose that function a is time independent. Let m ∈ N and S

= (

¹₂

− ,

¹₂

+ ) × { x } × {γ }. Define q

₀

( t ) = 1 and q

_j

( t ) = sin ( j π(

^t−¹²

+ 1 )) ∀ 1 ≤ j ≤ m . Then the functions ( q

_j

)

0≤j≤m

generate M ( S

) a vector space of dimension m + 1 . In addition, the functions q

_j

:

(i) are orthogonal with respect to both the L

²

( S

), L

²

( S

, a

^x

(., x , γ )) and L

²

( S

, a

^y

(., x , γ )) inner product,

(ii) verify

¹₂₊

1

2−

q

_k

( t ) q

_j

( t ) dt = 0 for all 0 ≤ k , j ≤ m where q

_k

( t ) denotes the derivative function of q

k

(t ).

When functions a

^x

, a

^y

are time dependent, we have:

Lemma 2 Let m ∈ N, suppose a

^x

(., ., .) (respectively a

^y

(., ., .)) to be C

¹

and bounded from below by positive constants, then there exists a family of regular functions (w

j

(t))

0≤j≤m

for t ∈ (

¹₂

− ,

¹₂

+ ) which generates M ( S

) a vector space of dimension m + 1 . In addition, the functions w

j

:

(i) are orthogonal with respect to L

²

( S

), L

²

( S

, a

^x

(., x, γ ) dt) and L

²

( S

, a

^y

(., x , γ ) dt ) inner products,

(ii) verify

¹₂₊

1

2−

(w

_k

( t ))w

j

( t ) dt = 0 for all 0 ≤ k , j ≤ m , where w

_k

( t ) denotes the derivative function of w

k

(t).

Proof Let ( q

_j

)

0≤j≤m

be the normalized basis derived from the L

²

− or t hogonal basis ( q

_j

)

0≤j≤m

. Since the function a

^x

(., ., .) (respectively a

^y

(., ., .) ) is bounded from below by a positive constant, the bilinear form

1 2+

1 2−

( q

_j

( t ))( q

_k

( t )) a

^x

( t , x , y ) dt

(respectively

¹₂+ 1

2−

( q

j

(t))( q

k

(t))a

^y

(t, x, y)dt) defines an L

²

-inner product. Let us denote

by M its matrix with respect to the basis ( q

_j

)

_0≤_j≤m

. The matrix M is symmetric positive

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defined, thus the spectral theorem claims that there exists an orthogonal basis with respect to L

²

-inner product denoted by (w

j

)

0≤j≤m

which diagonalizes the matrix M . The expression of the bilinear form in the basis (w

j

)

0≤j≤m

is diagonal, thus the vectors (w

j

)

0≤j≤m

are orthogonal to each others with respect to the bilinear form. Since the (w

_j

) functions are linear combinations of ( q

_j

) functions, the second items of Lemma 2 still holds true for the (w

_j

) functions. Lemma 2 is proved.

We consider, in the following, the general case where the speed depends on time. The time-independent case is treated by simply replacing the vectors w

k

by q

k

.

Define the 2-D velocity a =

a

^x

( t , x , y ) a

^y

( t , x , y )

and the incoming part of the boundary ∂ S

2₋

= {(x, y), (a/n) < 0}, where n is the outward normal to S

2

.

Definition 1 The decomposed problem associated to the problem (1) consists in finding ( u

₁

, u

₂

) such that:

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩

(u

1

, u

2

) ∈ H (β, Q

1

) × M( S

) ⊗

0

H

¹

((

¹₂

− η,

¹₂

+ η) × (γ, 1)), u

₂

=

^m

k=0

w

k

( t ) u

_2k

( x , y ) and (β/∇ u

₁

) = f i n Q

₁

,

u

₁

= 0 on ∂ Q

₁₋

,

(β/∇ u

₂

) = f i n Q

₂

,

b

1

(u

1

− u

2

, w

k

) =

¹₂+ 1

2−

(u

1

(t, x , γ )

−u

2

(t, x , γ ))q

k

(t)a

^y

(t, x , γ ) dt = 0, ∀x ∈ (

¹₂

− η,

¹₂

+ η), ∀q

k

∈ M(S

).

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The last equation of this system is the boundary condition on the incoming part of the domain S

2

in L

²

(

¹₂

− η,

¹₂

+ η).

Let u be the solution of the problem (1) and u

₁

its restriction on Q

₁

. We introduce u

₂

=

_m

k=0

w

k

( t ) u

_2k

( x ). Then u

₂

is the solution restricted to Q

₂

and u

₂

∈ M ( S

) ⊗

0

H

¹

((

¹₂

− η,

¹₂

+ η) × (γ, 1 )) . The decomposed problem is obtained by multiplying (1) by w

k

(t) ∈ M( S

) and by integrating with respect to t. Observe that

1 2+

1 2−

∂ u

₂

∂ t w

k

( t ) dt = 0 ∀w

k

∈ M ( S

);

and obviously the following condition:

b ( u

₁

− u

₂

, w

k

) =

1 2+

1 2−

( u

₁

( t , x , γ ) − u

₂

( t , x , γ ))w

k

( t ) a

^y

( t , x , γ ) dt = 0

yields the boundary condition.

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Theorem 2 The decomposed problem given in definition 1 is equivalent to the following reduced problem which consists in finding ( u

₁

, u

₂

) such that:

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩

(u

1

, u

2

) ∈ H(β, Q

1

) × M( S

) ⊗

0

H

¹

((

¹₂

− η,

¹₂

+ η)

×(γ, 1)) and (β/∇u

1

) = f i n Q

₁

,

u

₁

= 0 ∂ Q

₁₋

,

A

^x,k^∂u_∂_x²

+ A

^y,k^∂u_∂y²

=

¹₂₊

1

2−

f ( t , x , y )w

k

( t ) dt a . e . i n S

₂

,

¹₂+

1

2−

u

1

(t, x, γ )w

k

(t)a

^y

(t, x, γ )dt

= u

2k

(x , γ )

¹₂₊

1

2−

w

k

(t)w

k

(t)a

^y

(t, x , γ )dt a.e. i n (

¹₂

− η,

¹₂

+ η) and

∀w

k

∈ M ( S

).

(3)

where A

^x^,^k

=

¹

2+ 1

2−

a

^x

( t , x , y )w

k

( t )w

k

( t ) dt and A

^y^,^k

=

¹

2+ 1

2−

a

^y

( t , x , y )w

k

( t )w

k

( t ) dt . Proof We have

u

2

=

m k=0

w

k

(t)u

2k

(x , y).

Then almost everywhere in S

₂

we have:

∂u

2k

∂x

⎛

⎜ ⎜

⎝

1 2+

1 2−

a

^x

(t, x, y )w

k

(t)w

k

(t) dt

⎞

⎟ ⎟

⎠ + ∂u

2k

∂y

⎛

⎜ ⎜

⎝

1 2+

1 2−

a

^y

(t, x, y)w

k

(t)w

k

(t) dt

⎞

⎟ ⎟

⎠

=

1 2+

1 2−

f (t, x , y)w

k

(t) dt ∀w

k

∈ M (S

).

Corollary 1 The zero order approximation corresponding to the decomposed problem (3) is given by

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

(β/∇u

1

)

2

= f i n Q

1

,

u

1

= 0 ∂ Q

1₋

,

a

^x

(

¹₂

, x, y)

^∂u_∂x²⁰

+ a

^y

(

¹₂

, x , y)

^∂u_∂y²⁰

= f (

¹₂

, x , y) a.e. i n (

¹₂

− η,

¹₂

+ η) × (γ, 1), u

1

(

¹₂

, x , γ ) = u

20

(x, γ ) ∀x ∈ (

¹₂

− η,

¹₂

+ η).

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2.1 Algorithmes for numerical simulation

Now, let us come to the numerical approximation of ( u

₁

, u

₂

), the solution to the decom-

posed problem (3). To supplement the results outlined in [3], we will give in that follows

the construction of the algorithm requiered to the numerical simulations. An upwind finite

differences method is used for u

1

in Q

1

and for u

2

in Q

2

. Let M, N be two fixed positive

integers and 0 < <

¹₂

, 0 < η <

¹₂

, 0 < γ < 1 be three real numbers.

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2.1.1 The numerical approximation on Q

₁

The time and space steps are defined by t =

_N¹

, x =

_M¹

and y =

_M^γ

. Introduce the family of points ( t

_n

, x

_i

, y

_j

) such that t

_n

= n t , x

_i

= i x and y

_j

= j y . For all n = 0 , 1 , . . . , N ; i = 0 , 1 , . . . , M and j = 0 , 1 , . . . , M we pose u

ⁿ_i_,_j

u ( t

_n

, x

_i

, y

_j

), a

_i^x,n_,_j

a

^x

( t

_n

, x

_i

, y

_j

) and a

_i,j^y,n

a

^y

( t

_n

, x

_i

, y

_j

) then we obtain the following schema:

u

ⁿ⁺¹_i,j

− u

ⁿ_i,_j

t + a

_i,^x,n+1_j

u

ⁿ⁺¹_i,_j

− u

ⁿ⁺¹_i−1,j

x + a

_i,^y^,_jⁿ⁺¹

u

ⁿ⁺¹_i,j

− u

ⁿ⁺¹_i,_j−1

y = f (t

n+1

, x

i

, y

j

) which can be writen as

Θ

ⁿ⁺¹

U

ⁿ⁺¹

= U

ⁿ

+ F

ⁿ⁺¹

(5)

where Θ

ⁿ⁺¹

=

⎛

⎜ ⎜

⎝

A

ⁿ⁺¹₁

0 . . . . . . 0 B

₂ⁿ⁺¹

A

ⁿ₂⁺¹

... ...

0 B

₃ⁿ⁺¹

A

ⁿ⁺¹₃

... ...

... ... ... ... 0

0 . . . 0 B

ⁿ⁺¹_M

A

ⁿ⁺¹_M

⎞

⎟ ⎟

⎠ , U

ⁿ

=

⎛

⎜ ⎜

⎝ U

₁ⁿ

U

₂ⁿ

... ...

U

_Mⁿ

⎞

⎟ ⎟

⎠ ,

F

ⁿ⁺¹

=

⎛

⎜ ⎜

⎜ ⎝ F

₁ⁿ⁺¹

F

₂ⁿ⁺¹

... ...

F

_Mⁿ⁺¹

⎞

⎟ ⎟

⎟ ⎠ , U

ⁿ_j

=

⎛

⎜ ⎜

⎜ ⎝ U

_{1 j}ⁿ

U

_{2 j}ⁿ

... ...

U

_{M j}ⁿ

⎞

⎟ ⎟

⎟ ⎠

and F

ⁿ⁺¹_j

=

⎛

⎜ ⎜

⎜ ⎝ t f

_{1 j}ⁿ⁺¹

t f

_{2 j}ⁿ⁺¹

... ...

t f

_{M j}ⁿ⁺¹

⎞

⎟ ⎟

⎟ ⎠ .

In the other hand the matrices A

ⁿ⁺¹_l

= (o

^l_{i j}^,ⁿ⁺¹

)

1≤i,j≤M

for 1 ≤ l ≤ M and B

_hⁿ⁺¹

= ( b

_{i j}^h,n+1

)

1≤i,j≤M

for 2 ≤ h ≤ M are such that

o

^l,n+1_{i j}

=

⎧ ⎪

⎨

⎪ ⎩

1 +

_x^t

a

_il^x,n+1

+

_y^t

a

_il^y,n+1

i = j,

−

_x^t

a

_il^x,n+1

i = j − 1,

0 elsewhere

and b

^h,n+1_{i j}

=

−

_y^t

a

_{i h}^y,n+1

i = j,

0, elsewhere .

2.1.2 The numerical approximation on Q

₂

For Q

₂

the time step is given by

t =

²_N

and the space steps are defined by

η

x =

2η

M

and

_γ

y =

¹^−γ_M

. The nodes are given by x

i

=

¹₂

− η + i

_η

x and y

j

= γ + j

_γ

y. For k = 0, 1, . . . , m and i, j = 1, . . . , M; we put v

_i,j^k

v

^k

(x

i

, y

_j

) then, using upwind finite differences method, we can write

v

_i,j^k

− v

_i−1,j^k

η

x A

_i^x,k_,_j

+ v

_i,j^k

− v

_i,^k_j−1

y A

^y,k_i_,_j

= F

_i,j^k

, (6)

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where

A

^x,k_i_,_j

=

1 2+

1 2−

a

^x

( t , x

_i

, y

_j

)w

k

( t )w

k

( t ) dt ,

A

_i,^y,k_j

=

1 2+

1 2−

a

^y

(t, x

i

, y

j

)w

k

(t )w

k

(t)dt

and

F

_i,j^k

=

1 2+

1 2−

f (t, x

i

, y

j

)w

k

(t)dt.

The integrals A

^x,k_{i j}

, A

_{i j}^y^,^k

and F

_i^k_,_j

are computed by using a Gauss quadrature formula.

Then the schema (6) can be written as

B

^k

V

^k

= F

^k

such that the matrix B

^k

is defined:

B

^k

=

⎛

⎜ ⎜

⎜ ⎝

C

₁^k

0 . . . . 0 D

₂^k

C

^k₂

... ...

0 D

₃^k

C

₃^k

... ...

... ... ... ... 0 0 . . . 0 D

^k_M

C

_M^k

⎞

⎟ ⎟

⎟ ⎠

(7)

where C

_l^k

= ( c

^l,k_{r s}

)

1≤r,s≤M

and D

_l^k

= ( d

_{r s}^l,k

)

1≤r,s≤M

for 1 ≤ l ≤ M . In the other hand

c

^l,k_{r s}

=

⎧ ⎪

⎨

⎪ ⎩

η

x A

^x,k_r_,_l

+ y A

_r^y,k_,_l

r = s ,

− A

_r^x,k_,_l

y r = s − 1 ,

0 elsewhere

and d

_{r s}^l,k

=

−

_η

x A

_r,l^y,k

r = s,

0 elsewhere .

We denote

V

_l^k

=

⎛

⎜ ⎜

⎜ ⎝ v

^k_1,l

v

^k₂_,_l

... ...

v

^k_M,l

⎞

⎟ ⎟

⎟ ⎠

f or l = 1 , . . . , M ; V

^k

=

⎛

⎜ ⎜

⎝ V

₁^k

V

₂^k

... ...

V

_M^k

⎞

⎟ ⎟

⎠ , F

₁^k

=

⎛

⎜ ⎜

⎜ ⎝

−D

₁^k

V

₀^k

+

η

x

γ

y F

_1,1^k

η

x

γ

y F

₂^k_,₁

... ...

_η

x

_γ

y F

_M,1^k

⎞

⎟ ⎟

⎟ ⎠ ,

F

_l^k

=

⎛

⎜ ⎜

⎜ ⎝

_η

x

_γ

y F

_1,l^k

η

x

γ

y F

_2,l^k

... ...

η

x

γ

y F

_M^k_,_l

⎞

⎟ ⎟

⎟ ⎠

and F

^k

=

⎛

⎜ ⎜

⎝ F

₁^k

F

₂^k

... ...

F

_M^k

⎞

⎟ ⎟

⎠

for l = 2 , . . . , M .

(8)

We denote by ( U

ⁿ

, V

ⁿ

) a numerical approximation of the solution ( u

₁

, u

₂

) of the decomposed problem (3) on whole domain Q at time t

_n

. Then the component U

_{i j}ⁿ

of the matrix U

ⁿ

represents an approximation of u

₁

( t

_n

, x

_i

, y

_j

) at every node ( t

_n

, x

_i

, y

_j

) of the domain Q

1

and the component

_m

k=0

w

k

(t

n

)v

_i,^k_j

of the matrix V

ⁿ

represents an approximation of u

2

(t

n

, x

i

, y

j

) at every node (t

n

, x

i

, y

j

) of the domain Q

2

. The following proposition allows to summarize the previous algorithm which calculates an approximation of the solution (u

1

, u

₂

) of problem (3) on the whole domain Q .

Proposition 1 The following algorithm gives an approximation of the solution (u

1

, u

₂

) of the decomposed problem (3):

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩ i n Q

1

U

⁰

= 0,

t =

_N¹

, x =

_M¹

and y =

_M^γ

,

t

n

= nt f or n = 0, 1, 2, · · · , N ,

x

_i

= i x f or i = 0 , 1 , 2 , · · · , M , y

_j

= j y f or j = 0 , 1 , 2 , · · · , M ,

ⁿ⁺¹

U

ⁿ⁺¹

= U

ⁿ

+ F

ⁿ⁺¹

f or 0 ≤ n ≤ N − 1, on t he i nter f ace

V

_M^k

(x

i

, γ ) =

¹₂+

12−u1(t,xi,γ )wk(t)a^y(t,xi,γ )dt ¹₂+

1

2−wk(t)wk(t)a^y(t,xi,γ )dt

f or 1 ≤ i ≤ M and 0 ≤ k ≤ m, i n Q

2

t =

²_N

,

η

x =

^2η_M

and

γ

y =

^1−γ_M

,

t

_n

=

¹₂

− + n

t f or n = 0 , · · · , N , x

_i

=

¹₂

− η + i

η

x f or i = 0 , · · · , M , y

j

= γ + j

γ

y f or j = 0, · · · , M,

B

^k

V

^k

= F

^k

f or 0 ≤ k ≤ m ,

V =

^m

k=0

w

k

( t

_n

) V

^k

f or n = 0 , 1 , · · · , N .

Conclusion As we can observe in the following simulations considered at final instant, the classical finite differences method doesn’t work for thin domains representing branches.

Besides, the variational reduction method gives a good description of the evolution of concentrations.

3 Numerical simulations

In what follows, numerical experiments are presented taking a

^x

( t , x , y ) = a

^y

( t , x , y ) = 50x + y + exp(0.1t), f (t, x , y) = (2y + x exp(0.5t))t, γ =

¹₂

, N = 20 and M = 200.

Only w

0

and w

1

are considered in M(S

). They are calculated thanks to Gramm-Schmidt process. We present in the Fig. 2 numerical simulations of the zero order model, the transport equation by finite differences method in the whole domain Q and the model obtained by the variational reduction strategy. Different values of the parameters η and are tested.

It can be checked that the finite differences method does not give accurate results for too

small values of and η compared to the zero order model which is the reference one. In the

other hand the proposed variatinal reduction method gives a similar simulation to the zero

order model and is not sensitive to the values of these parameters.

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Fig. 2 The numerical representation

Acknowledgments This work has been supported with a grant PHC Volubilis from the French foreign office and the marocain ministry of education and research MA/11/246.

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