## Slow/fast kinetic scheme with slow diffusion

### P. Rouchon

### Centre Automatique et Syst`emes, ´ Ecole des Mines de Paris 60 Bd. Saint Michel, 75272 Paris cedex 06, FRANCE.

### Email: [email protected].

### Internal Report A195 March 1997

This short note follows [3]. We consider here slow/fast chemical reactions with additional slow diffusions:

*∂x*

*∂t* =*η∆x*+*v(x, ²)* (1)

where *x*= (x1*, . . . , x**n*) are the concentration profiles, ∆ =*∇ · ∇,* *²*is a small
positive parameter, *η* (∼ *²) is the diffusion matrix (symmetric and positive*
definite) and *v(x, ²) corresponds to the kinetic scheme. As in [3], we assume*
the following slow/fast structure for*v* [4]:

A1 for *²*= 0, dx

dt =*v(x,*0) admits an equilibrium manifold of dimension *n**s*,
0*< n**s**< n, denoted by Σ*0.

A2 for all*x*0*∈*Σ0, the Jacobian matrix, *∂v*

*∂x*

¯¯

¯¯

(x0*,0)*

admits*n**f* =*n−n**s*eigen-
values with a strictly negative real part (the eigenvalues are counted with
their multiplicities).

Locally around Σ0, there exists a partition of*x*into two groups of components,
*x*= (x*s**, x**f*), with dim(x*s*) =*n**s* and dim(x*f*) =*n**f*, such that the projection
of Σ0 on the*x**s*-coordinates is a local diffeomorphism.

Using the approximation lemma of invariant manifold, and its version for
slow/fast systems [2, theorem 5, page 32], we are looking for an asymptotic
expansion versus *²,*

*x** _{f}* =

*h*

_{0}+

*h*

_{1}+

*. . . ,*

1

of an invariant slow manifold of (1) closed to Σ0 (h0, *h*1 , . . . depend on the
profiles *x**s*). The slow equations derived here below, and, in particular, the
slow diffusion terms, admit a rather unexpected form that has, as far as we
known, nether been derived elsewhere.

Following [2, 3], the zero order approximation*h*0is defined by the algebraic
equation (in the sequel, we do not recall the dependence versus*²)*

*v**f*(x*s**, h*0) = 0.

*h*1is obtained by zeroing of the first order term in

*∂x**f*

*∂t* *−*
µ*∂h*0

*∂x**s*

+*∂h*1

*∂x**s*

+*. . .*

¶*∂x**s*

*∂t* = 0.

Using the shortcut notations *h*0,s= *∂h*0

*∂x**s*

, . . . we have

(v*f,f**−h*0,s*v**s,f*)h1=*h*0,s(η*s*∆x*s*+*η**sf*∆h0+*v**s*)*−η**f s*∆x*s**−η**f*∆h0

with

*η*=

µ *η**s* *η**sf*

*η**f s* *η**f*

¶

*,* ∆h0=*h*0,ss(∇x*s**,∇x**s*) +*h*0,s∆x*s*

and where the functions are evaluated at (x_{s}*, x** _{f}*=

*h*

_{0}(x

*)). Using*

_{s}*h*0,s=

*−v*

_{f,f}

^{−1}*v*

*f,s*

*,*

we obtain the following first order approximation of the slow dynamics:

*∂x**s*

*∂t* = *C(x**s**, h*0)(η*s*∆x*s*+*η**sf*∆h0+*v**s*(x*s**, h*0))

+E(x*s**, h*0)(η*f s*∆x*s*+*η**f f*∆h0) (2)
where the correction matrix *C*is identical to the one in [3],

*C*= 1*−v**s,f*(v_{f,f}^{2} +*v**f,s**v**s,f*)^{−1}*v**f,s*

and*E* is defined by

*E*=*−v**s,f*(v_{f,f}^{2} +*v**f,s**v**s,f*)^{−1}*v**f,f**.*

When the kinetics *v* is in Tikhonov normal form, i.e., *v* = (²v*s**, v**f*), we
recover (up to second order terms in*²) the classical reduced model*

*∂x**s*

*∂t* =*η**s*∆x*s*+*η**sf*∆h0+*v**s*(x*s**, h*0), *v**f*(x*s**, h*0) = 0.

2

We will consider now the case, already pointed out in [1] and considered in [3], of affine fast fibers: the change of coordinates yielding to Tikhonov normal form is linear. Using notation of [3, section 4, equation (18) ], (1) admits the special structure

*∂x**s*

*∂t* = *η**s*∆x*s*+*η**sf*∆x*f*+*A**ss* *²˜v**s*(x*s**, x**f*) +*A**sf* *v*˜*f*(x*s**, x**f*)

*∂x**f*

*∂t* = *η**f s*∆x*s*+*η**f*∆x*f*+*A**f s**²˜v**s*(x*s**, x**f*) +*A**f f* *v*˜*f*(x*s**, x**f*).

The change of coordinates

(x*s**, x**f*)*7→*(ξ=*x**s**−A**sf*(A*f f*)^{−1}*x**f* *, x**f*).

leads to

*∂ξ*

*∂t* = (η*s**−A**sf**A*^{−1}_{f f}*η**f s*)∆x*s*+ (η*sf* *−A**sf**A*^{−1}_{f f}*η**f f*)∆x*f*

+(A*ss**−A**sf*(A*f f*)^{−1}*A**f s*)*²˜v**s*

*∂x**f*

*∂t* = *η**f s*∆x*s*+*η**f*∆x*f*+*A**f s**²˜v**s*+*A**f f*˜*v**f**.*
Assuming that eigenvalues of

*A**f f*

³

*v**f,f*+*A**sf**A*^{−1}_{f f}*v**f,s*

´

have strictly negative real parts, then the quasi-steady-state method can be applied and leads to the following slow system

*∂ξ*

*∂t* = (η*s**−A**sf**A*^{−1}_{f f}*η**f s*)∆x*s*+ (η*sf* *−A**sf**A*^{−1}_{f f}*η**f f*)∆x*f*+*. . .*
*. . .*+ (A*ss**−A**sf*(A*f f*)^{−1}*A**f s*)*²˜v**s*

0 = ˜*v**f**.*

Pulling back into the original coordinates (x*s**, x**f*) yields:

*∂x**s*

*∂t* =
h

1*−A**ss*(A*f f*)^{−1}*v*^{−1}_{f,f}*v**f,s*

i* _{−1}* ³

(η*s**−A**sf**A*^{−1}_{f f}*η**f s*)∆x*s*+*. . .*
*. . .*+ (η*sf* *−A**sf**A*^{−1}_{f f}*η**f f*)∆x*f*+ (A*ss**−A**sf*(A*f f*)^{−1}*A**f s*)*²˜v**s*

´
0 = ˜*v**f*(x*s**, x**f*).

Let us finish with a small example derived from [3, equation (1)] by adding slow diffusion:

*∂x*1

*∂t* = *η*1∆x1*−k*1*x*1+*k*2*x*2*−²kx*1*x*2

*∂x*2

*∂t* = *η*2∆x2+*k*1*x*1*−k*2*x*2*.* (3)
3

Setting *∂x*2

*∂t* to zero et neglecting *η*2 yields an incorrect slow model (diffusion
and kinetics)

*∂x*1

*∂t* =*η*1∆x1*−²(kk*1*/k*2)x^{2}_{1}

whereas reduction via the above computations provides the correct slow equa- tion

(1 +*k*1*/k*2)*∂x*1

*∂t* = (η1+*η*2*k*1*/k*2)∆x1*−²(kk*1*/k*2)x^{2}_{1}*.*

### References

[1] V. Van Breusegem and G. Bastin. A singular perturbation approach to the reduced order dynamical modelling of reaction systems. Technical Report 92.14, CESAME, University of Louvain-la-Neuve, Belgium, 1993.

[2] J. Carr. *Application of Center Manifold Theory. Springer, 1981.*

[3] P. Duchˆene and P. Rouchon. Kinetic scheme reduction via geometric sin-
gular perturbation techniques. *Chem. Eng. Science, 51:4661–4672, 1996.*

[4] N. Fenichel. Geometric singular perturbation theory for ordinary differential
equations. *J. Diff. Equations, 31:53–98, 1979.*

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