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Maxence BIGERELLE, Alain IOST - Physical Interpretations of the Numerical Instabilities in
Diffusion Equations Via Statistical Thermodynamics - International Journal of Nonlinear Sciences
and Numerical Simulation - Vol. 5, n°2, p.121-134 - 2004
©Freund Publishing House Ltd. International Journal of Nonlinear Sciences and Numerical Simulation 5(2), 121-134, 2004
Physical Interpretations of the Numerical Instabilities in
Diffusion Equations Via Statistical Thermodynamics
M. Bigerelle and A. lost
Laboratoire de Metallurgie Physique et Genie des Materiaux (UST Lille),
Equipe Surfaces et Interfaces (ENSAM Lille) - CNRS UMR 8517.
ENSAM Lille, 8 boulevard Louis XIV, 59046 LILLE Cedex, FRANCE
Abstract
The aim of this paper is to analyze the physical meaning of the numerical instabilities of the parabolic partial differential equations when solved by finite differences. Even though the explicit scheme used to solve the equations is physically well posed, mathematical instabilities can occur as a consequence of the iteration errors if the discretisation space and the discretisation time satisfy the stability criterion. To analyze the physical meaning of these instabilities, the system is divided in sub-systems on which a Brownian motion takes place. The Brownian motion has on average some mathematical properties that can be analytically solved using a simple diffusion equation. Thanks to this mesoscopic discretisation, we could prove that for each half sub-cell the equality stability criterion corresponds to an inversion of the particle flux and a decrease in the cell entropy in keeping with time as criterion increases. As a consequence, all stability criteria defined in literature can be used to define a physical continuous 'time-length' frontier on which mesoscopic and microscopic models join.
Keywords:
D i f f u s i o n , Partial D i f f e r e n t i a l E q u a t i o n s , Finite D i f f e r e n c e M e t h o d , Stability, Fractal, M o n t e C a r l o , Entropy, Anti E n t r o p y P r o c e s s , B r o w n i a n M o t i o n .1 Introduction
The rapid d e v e l o p m e n t of h i g h - s p e e d c o m p u t e r technology over the recent y e a r s h a s been a c c o m p a n i e d by a very substantial g r o w t h of computational science. A high n u m b e r of physical p h e n o m e n a can be m o d e l e d by Partial Differential E q u a t i o n s ( P D E ) [1] that can be solved by n u m e r o u s numerical m e t h o d s [2-3], A m o n g these m e t h o d s , only the Finite D i f f e r e n c e M e t h o d ( F D M ) [4], w h i c h s t a n d s out as being universally a p p r o p r i a t e to both linear and non-linear p r o b l e m s , is considered here. F D M can equally be applied to hyperbolic, elliptic and parabolic equations. In short, these equations are used to model w a v e physical properties, time-stationary p h e n o m e n a and all time d e p e n d e n t d i f f u s i o n (transport) processes. T o solve these P D E numerically, F D M consists
in e x p a n d i n g all v a r i a b l e s of t h e P D E in T a y l o r series on a grid at d i f f e r e n t points in the relevant d o m a i n , w h i l e limiting t h i s e x p a n s i o n to the first derivatives, i n t r o d u c i n g t h e s e finite e x p a n s i o n s on the P D E p r o b l e m a n d f i n a l l y solving it by a d e q u a t e n u m e r i c a l m o d e l s . T h e main problem e n c o u n t e r e d is that P D E b e c o m e s a discretised equation in w h i c h all d i f f e r e n t i a l elements do not tend t o w a r d s z e r o but t o w a r d s fixed v a l u e s that d e p e n d on t h e n u m b e r of discretised points. H o w e v e r , t h e d i s c r e t i s a t i o n process involves local truncation e r r o r s and the problem of t h e consistency of t h e n u m e r i c a l s c h e m e is raised, i.e. the solution c o n v e r g e s t o w a r d s the solution of P D E as the mesh length t e n d s t o w a r d s zero. S u p p o s i n g that t h e n u m e r i c a l s c h e m e is consistent, a s e c o n d p r o b l e m consists in testing the stability of this s c h e m e since all r o u n d i n g errors introduced d u r i n g c o m p u t a t i o n (as a
122 Μ . B i g e r e l l e a n d A. l o s t / I n t . J . N o n l . Sei. N u m . S i m u l a t i o n 5 ( 2 ) , 1 2 1 - 1 3 4 , 2 0 0 4
c o n s e q u e n c e of the finite representation n u m b e r s ) increase d u r i n g the iterative s c h e m e itself that may b e c o m e unstable. M a n y mathematical t o o l s and t h e o r e m s can be used to study the stability o f t h e system. H o w e v e r no physical interpretation h a s been p r o p o s e d yet. These m e t h o d s s h o w that by increasing the discretisation time, the system m a y p a s s f r o m stability to instability. In fact, these mathematical results involve q u e s t i o n s about the physical m o d e l i n g process. T h e r e exists a space-time relation that a l l o w s u s t o solve the p r o b l e m of consistence related to C h a o s T h e o r y : a small initial error m a y be raised and b e c o m e very important leading to a total instability of the system [5], H o w e v e r P D E model m a c r o physical systems and the f u n d a m e n t a l questions are: ' I s the length of t h e s p a c e - t i m e unit used in discretisation c o m p a t i b l e with t h e m i c r o s c o p i c p h e n o m e n a ? If not d o e s it lead to an unstable s y s t e m ? ' With regard to the high n u m b e r of mathematical p u b l i c a t i o n s related to P D E stability, w e p r o p o s e to study physically the stability of a parabolic P D E that can be applied onto physical p r o b l e m s w h i c h are easily modeled at the m i c r o s c o p i c scale.
In a first part o f t h e p r e s e n t paper, w e shall describe t h e physical use of these equations. Secondly, w e shall a n a l y z e t h e numerical solution of the w e l l - k n o w n Fick equation [6] compared with t h e Einstein model [7] to find out if the problem is well posed. In a third part, we shall introduce briefly a n e w method for studying stability. In a fourth part, w e shall propose a physical interpretation of stability based on B r o w n i a n motion theories [5,7] and the second principle o f t h e r m o d y n a m i c s [8]. Finally w e shall proceed a M o n t e Carlo simulation to illustrate our physical stability interpretation.
2 Parabolic Differential Equations
The Parabolic P D E is given by the following mathematical definition:
du ^ d d r i \ . . .
ot ~7 ox, ox i
w h e r e ( χ , / ) ε Ω χ IR+, a n d Ω is a n o p e n set o f
I Rn.
T h e s e e q u a t i o n s that c h a r a c t e r i z e transport p h e n o m e n a such as heat t r a n s f e r , viscous fluid m e c h a n i c s , a t o m - v a c a n c y transport, O h m law, c o n t a g i o u s e p i d e m i c ... can be obtained by three main w a y s :
(1) C o u n t i n g the n u m b e r o f " p a r t i c l e s " by v o l u m e unit and a p p l y i n g a Fick law f l u x J cc grad(u);
(2) U s i n g transport e q u a t i o n s f o r each entity of the system (particle) and m a k i n g t h e f r e e path tend t o w a r d s zero;
(3) U s i n g stochastic p r o c e s s e s as B r o w n i a n motion.
T h e s e mathematical m e t h o d s , used for physical modeling, s h o w that the d i f f u s i o n process can be derived f r o m probabilistic topics, m e a n i n g that d i f f u s i o n laws are stochastic in nature. A s y m p t o t i c c o n s i d e r a t i o n s will lead to suppressing t h e stochastic aspect of these m o d e l s to obtain d e t e r m i n i s t i c ones. In other terms, m i c r o s c o p i c f l u c t u a t i o n s are " r e m o v e d " to result in m a c r o s c o p i c f o r m u l a t i o n s , i.e. c o n t i n u o u s o n e s . O n e important c o n s e q u e n c e o f this f u n d a m e n t a l duality is that the differential elements can not have a z e r o limit that w o u l d lead to a n o n s e n s i c a l interpretation. For e x a m p l e , in the Fourier heat t r a n s f e r law, heat can be considered a s the B r o w n i a n motions of p h o n o n s / e l e c t r o n s and then the differential elements c a n n o t be smaller than the mean f r e e path of the m o t i o n . O n the contrary, the differential e l e m e n t s must not be too large to include the m e s o s c o p i c m o d e l i n g of the p h e n o m e n o n . T h e s e c o n s i d e r a t i o n s show that there exists a scale range w h e r e the differential elements get a physical meaning, w h o s e determination is no trivial w o r k .
3. The resolution of the diffusion equation
3.1. Analytical solution
Equation 1 can be analytically solved for only specific systems, and t h e r e f o r e a numerical estimation has to be generally processed. In this part Eq.l is discretised by the Finite D i f f e r e n c e Method. W i t h o u t lack of generality, w e shall retain to treat in this paper the simple m o n o
-ISSN 1565-1339 International Journal o f N o n l i n e a r Sciences and N u m e r i c a l Simulation 5(2), 121-134, 2 0 0 4 123
dimensional Fick equation that is reduced to:
(2)
dc{x,t) Dd2C{x,t)dt dx2
where C(x,t) is the concentration at time t of particles at position χ , and D the diffusion coefficient. With C(o,o)= C0S0, w h e r e δ0 is the Dirac pulse, and as initial conditions, the well-known solution is:
Cn f
ADt
C(x,/) = - r e x p (3)
2 yfnDt
We shall precise some of its properties. Firstly, Eq.3 reduced by C0 is a Gaussian Probability
Density Function ( P D F ) with zero mean and
a(t)=y[2Dt standard deviation. This clearly
means that diffusion is a stochastic process and that equations can be obtained by asymptotic considerations (like the Central Limit Theorem). To visualize this fact, a M o n t e Carlo method will be used in chapter 5 to simulate the diffusion process on purpose to compare with the theoretical solution given by Eq.3.
3.2. Numerical solution
We shall now use the finite difference method to discretise Eq.2. Using Taylor expansions:
dC(x,t) _ C(iAx, ( j + l)A?)-C(/'Ax, /At) dt x=iAxJ = /Al At + Θ(Δί) = — — + Θ(Δί), At and d2C(x,t) dx2 r> - 2 C + CJ (Ax)2 +l +Θ((ΔΧ)2) χ=ιΔχ.ι=/&ι
which gives the following explicit numerical scheme:
DAt
crx = a • • ( c / _ , - 2 C / + C /+ 1) . (4) On the other hand, if the gradient is considered at time t = ( j + \)At, then the f o l l o w i n g implicit numerical scheme is obtained:
dC(x,t) dt . Dd2c(x,t) 1 — Γ j χ=/Δχ,/=J&I DAt dx2 x=iAx,l=(j+\)Al (Δχ) (5) Or more generally: (Axf (i
(6)
which corresponds to the well-known Crank-Nicholson s c h e m e for β - 0.5.
Equation 4 is an explicit scheme m e a n i n g that concentration at time t + At only d e p e n d s on the gradient at time t . Equation 5 is an implicit scheme m e a n i n g that concentration at time
t + At only d e p e n d s on the gradient at time t + At . N e g l e c t i n g problems of discretisation
errors (which will be minimized w h e n β = 0.5 in Eq.6.) and numerical stability, the question which arises f r o m our analysis is to k n o w whether the s c h e m e gets a physical sense or not.
3.3. The implicit/explicit scheme and the Einstein theory of Brownian motion
T o a n s w e r the above question, w e shall analyze the hypothesis used by Einstein in 1905 to give a physical explanation to the phenomenological coefficient D in Eq.2 introduced by Fick. Einstein postulated that concentration at t i m e t + At is given by:
C(x, ί + Αί)=Σ W(X,, At)p{x - X,, t), ( 7 ) isp
where w(x,At) is the probability that an i particle makes a Xi long j u m p during a time At. This equation is reduced to Eq.2 by posing
D = Htri jlAt). A s a consequence, Einstein
shows that the particle concentration at time
t + At only d e p e n d s on average on the
concentration at t i m e t , i.e. before the j u m p occurs. T h e implicit scheme ( β = 0 ) lets us suppose an instantaneous j u m p , which is a physical non-sense.
The main criticism against the implicit s c h e m e is that w e cannot a f f i r m that all p a r t i c l e s w o u l d j u m p s i m u l t a n e o u s l y at t i m e t, w h i c h . i s i n
f a c t well f o u n d e d . H o w e v e r this problem is still under discussion: in a discretised space, the mesoscopic time At ( Δ / » Γ_ Ι w h e r e Γ is the
atom j u m p f r e q u e n c y ) could be used without any physical problem to model the d i f f u s i o n states without taking into account the correlation effects, and then the particles can j u m p
124 Μ. Bigereile and A. l o s t / I n t . J . Nonl. Sei. Num. Simulation 5(2), 121-134, 2004
simultaneously. However, even if time is
mesoscopic, Eq.7 can be transformed into:
C(x, t + \t)=Y
jW(X
i,At- At, )c(x-X„t- At,) ,(8)
iepand At, ( At, «At ) will be seen as time
fluctuation. By introducing this perturbation in
the probability jump, the variance properties of
the concentration become less smooth than the
average solution. This result was reported by El
Naschie in his study of the abnormal diffusion
and correlation in the Cantorian space [9], The
fractal space-time cannot exactly be d=2, but
must be slightly larger because diffusion on a
lattice and in a random environment is harder
than in a smooth space. Therefore the fractal
dimension must be taken as the mean <d
c>=2
rather than d
c=2. As a consequence, introducing
a perturbation in the Einstein diffusion equation
will not affect the solution on average, like the
mesoscopic diffusion problems in the EDP
theory. However the Fractal concepts of the
Brownian motion show that the positions of the
particles x[t) possess the property
(t + At)-X(t)f^K At and the graph X(t)
versus time possesses a statistical self-affine
structure meaning that
(x{t + At)-X(t)) = r-
05((x(t + rAt)-X(t))) As a
consequence, time and distance do not possess a
Euclidean metric and then to use a Euclidean
formulation (integer integration of PDE), as
used by Einstein, the Brownian Motion must be
located at a constant time At involving a
constant displacement X . Therefore, in the
average dimension analysis, the formulation of
Eq.8 is rejected for Einstein's Eq.7. All these
remarks show that only the explicit formulation
gets a physical meaning even though Eqs.(5-6)
introduce some physical non-sense.
4. Stability study
4 . 1 . T h e s t a b i l i t y - i n s t a b i l i t y t r a n s i t i o n
We have shown that the explicit scheme
gets a physical sense that the implicit one does
not possess. As concentration at time t + At is
calculated from concentration at time t , the
explicit scheme becomes a dynamic system
given by Eq.5 that may become unstable for a
25000 50000 75000 1ES 0 25000 50000 75000 1E5 25000 50000 75000 1ES
(/) 0 25000 50000 75000 I £5 25000 50000 75000 1E5 25000 50000 75000 tE5
A
eeuJ
λ=0,50018 ^ •2E11 λ=0,50019 -βεπ 25000 50000 75000 1ES 0 25000 50000 75000 tES Number of iterations 25000 50000 75000 1E5Fig. 1. Numerical estimation of a PDE problem
in the middle of the grid C
A(50,t) versus the
number of iterations used in the numerical
scheme for different values of the stability
criterion λ . 20 simulations are computed with
different initial noises. The analytical solutions
are limC(50,?)= 100.
given set of values DAt/(Axf . The essential
idea behind stability is that a numerical process,
when applied wisely, should limit the
amplification of all components of the initial
conditions including rounding errors. The basic
analysis considers the growth of perturbations in
initial data or the growth of errors introduced at
mesh points at a given time level. To us, this
mathematical philosophy seems close to the
Chaos Theory philosophy and as a consequence
we postulate that a duality exists between the
mathematical interpretation and the physical
representation. Three common methods exist to
investigate stability: the Von Neumann method,
the matrix method, and the energy method [4],
Using one of these three methods, it can be
shown [4] that the explicit scheme will be stable
if:
, DAt 1
< —
M - 2 · <
10>
A stochastic simulation will be used to illustrate
the PDE stability. Let us take the following
parameters to process the numerical simulation:
D=\ , Δχ = 1 , C(0,i)=100 , C(l00,i)=100 and
C(x,0)= U,xe {l...99}where U is a random value
that follows a uniform law in the range
ISSN 1565-1339 International Journal o f N o n l i n e a r Sciences and Numerical Simulation 5(2), 121-134, 2 0 0 4 125
£/e[-0.5· 0.5] used to introduce a perturbation in the numerical scheme. The analytical solution gives limC(x,r) = 100 , Vx . Then the explicit
/-» oo
scheme is used to calculate the values of Cx(x,t)
at different times and space discretised points. For sufficiently large values of t (t = tx =100000) we can admit that the solution
is reached since convergence errors are lower than computer errors due to finite representation numbers. To analyze stability, we shall compute the estimation values Cx(50,t) , the
concentration in the middle of the medium at different times for different values of A
( A e [0.40000, 0.50020] ). Fig. 1 is the plot of
(Γλ(50,/) versus time for different values of A
(for a given Α , 20 simulations are computed with different initial noises U ). For A < 0.5005 , the initial noise decreases during iterations and the numerical solution C/t(50,i) converges
towards the true value i.e. 100. This threshold is slightly higher than the 0.5 theoretical value, which is not in contradiction with the stability theory. In fact, this theory affirms only that a numerical scheme will be unconditionally stable for A < 0.5, i.e. whatever the initial conditions, the frontier conditions, At , Ax or D , the scheme will always be stable if A <0.5 . The theory does not affirm that some particular conditions can lead to stability for Ac > A > 0.5 .
For example, with Δχ = 10 , Ac =0.512 and
Δχ = 5 leads to Ac = 0.503 (in fact, we have
shown that if all parameters stay unchanged, lim Ac = 0.5). Fig. 1 shows that for A > 0.5005, a
noise appears on the values Cx (50,/)= 100 that
will be exponentially amplified with an increase of A : the scheme becomes unstable. Then, we estimate the error by:
Er{xl2,0= log | C , ( x / 2 , O - 1 0 0 | \
Fig.2 represents this error function versus A and confirms that after A > 0.50005 , error increases exponentially according to the power-law:
4.2. Space-time dimension and stability process Before giving a physical interpretation, five
bU 40 Τ 30 2 0 t t Τ 30 2 0 10 t 10 0 : Ϊ t -10 -20 * * -10 -20 * -30 : --30 -40 ο in ο in ο in ο α) σ> ο ο t- CM σι σ> ο ο ο ο ο σ> σ> ο ο ο ο ο -*r m m in in m ö ö d ö ö ö 0 Stability Criterion
Fig. 2. Numerical estimation of the error for a PDE problem in the middle of the grid CA(50,i) versus the number of the iterations used in the numerical scheme for different values of the stability criterion A. 20 simulations are computed with different initial noises and means on CA(50,f). The associated 95% confidence levels are plotted.
remarks about inequality (10) have to be made: 1) For given values of D and At, if the space increment Δχ decreases, then after a critical value, the scheme becomes unstable. In other words, the time At is too large to isolate the diffusion phenomenon on a length Δχ . In the same way, if At decreases Δχ must decrease, i.e. the length must be small enough to isolate the diffusion flux during At time which looks like a kind of macroscopic uncertainty principle. 2) Since diffusion can be seen as a random process X(t) with Gaussian increment we obtain:
v a r [ x { t
2) - X ( t ^ \ t
2- t \ ^
D' , (11)
where Df = 1.5 is the average fractal dimension
of the Brownian motion graph and var(X) the variance of X. A characteristic length Δχ of the displacement can be given by the standard deviation of E q . l l and then
yl\ar[x(t2)-x(tt)] = Δχ . Therefore At=\t1-t\
can be seen as the characteristic length associated with At time and we obtain
Ax <x VÄ7 or At/(Ax)2=cte that represents a limit
condition between stability and instability. As a consequence, the criterion of stability follows the same relation as the relation between time and distance in the fractal concept of diffusion.
126 Μ. Bigereile and A. l o s t / I n t . J . N o n l . Sei. N u m . Simulation 5(2), 121-134, 2 0 0 4 STABLE Macroscopic System Particle I I I 1 Γ I I 1 1
2D
MEASURABLEVelocity
and interaction \
particles Dc=rtmlh
UNSTABLEno interaction particle
1 I A2 7tm
UNMEASURABLE ->At
(A,)2 Table 1 Another stability criterion (with different powerexponents) will be incompatible with the space-time relation for the fractal motion of a particle. 3) Let us analyze the solution of a sandwich of particles that diffuse in an infinite medium. The solution is given by Eq.3. This solution is a Gaussian PDF (at a constant factor) with a zero mean and a σ(/) = ~j2Dt standard deviation. According to the properties of the Gaussian PDF, 66% of the diffused mass are in the interval [ - σ ( 4 σ(ί)], 95% in [ - 2σ(ί), 2σ(ί)}, 99.7% in
[- 3σ(ή, 3σ{ί)]... This clearly means that the
value ψ ( ρ ) σ ( / ) with: ψ</<) ,
( χι\
-JTJC
e x p
dx = P
(12)-T </>)
can be seen as a mean diffusion front (without any fractal aspect) and as a consequence
L{P) = Ψ(Ρ) λ/2 Dt becomes a characteristic
length of the diffusion progression. Considering now each discretised space length Δλ , an elementary sandwich diffusion process centered on Ax interval must statistically keep a certain mass on the Δχ interval, then Ψ(Ρ) -J2Dt < Ax and finally:
DAt V{P)
W 2 • <l3>
This condition looks like the stability condition given by Eq.10. It physically means that stability can be obtained in a macroscopic system only if the adimensional parameter DAt/(Ax)2 is lower
than a characteristic length related to the
diffusion process. It must be emphasized that the main relation of the thermodynamic interpretation of this instability proposed in the following paragraph exactly leads to the stability conditions of the EDP when posing
T(/>)=1, as a result of the Gaussian properties
related to the inflexion point.
4) From the Heisenberg principle, AxAp>h and then At/ Ax2 < h/2Trm. This relation looks like a
stability criterion. As 1/2D < \ / 2 Dc = hl2mn
(meaning that D>Dc=mnlh ), Dc is the
smallest Diffusion Coefficient (without any environmental interactions) that can be measured according to the uncertainty principle and which leads to a stable scheme.
If At/(Ax)2 e [l/2D, A/2jun] (see table 1) then the
scheme is unstable but the particle flux can be measured. D is a macroscopic coefficient that represents the easiness with which a number of atoms passes through a fictive interface by time unit. This means that D involves a correlation effect, interactions between atoms, a potential barrier not involved in Dc (Heisenberg Principle
concerns isolated particles). In the range [D, Z)c],
all is seen if measurable instability is due to the interaction between particles, velocity ... which seems to converge to the fact that chaos will appear when different systems interact and these interactions are related to the coefficient D. 5) If D increases for a constant value of Δ//(&χ:)2
(constant space-time observation), the system might become unstable. Increasing D corresponds to increasing the diffusion front
ISSN 1565-1339 International Journal of Nonlinear Sciences and Numerical Simulation 5(2), 121-134, 2004 127 length in t h e e l e m e n t a r y cell a n d t h e n f r o m ( 1 3 ) t h e s y s t e m m a y b e c o m e u n s t a b l e . W i t h a s a m e v a l u e of s p a c e - t i m e o b s e r v a t i o n , t h e m a p m i g h t b e c o m e u n s t a b l e w h e n t h e v e l o c i t y o f t h e particle i n c r e a s e s (e.g. i n c r e a s i n g t e m p e r a t u r e increases D ) w h i c h is a w e l l - k n o w n e x p r e s s i o n of the e m e r g e n c e o f c h a o s in t h e f l u i d as o b s e r v e d in the c l a s s i c a l T a y l o r - C o u e t t e e x p e r i m e n t a t i o n [10].
5. Physical interpretation of
stability
W e shall n o w i n t r o d u c e a p h y s i c a l interpretation of stability in w h i c h w e s u p p o s e that Eq.2 g o v e r n s t h e d i f f u s i v e s y s t e m . T h i s e q u a t i o n can b e s e e n as m o d e l i n g t h e d i f f u s i o n in all t h e m e d i a a n d t h e r e f o r e is an i n t r i n s i c physical p r o p e r t y o f t h e s y s t e m . In t h i s s y s t e m , w e shall i n t r o d u c e t h e initial c o n c e n t r a t i o nc(x,0), a space description Ω (open set of IR), and limit c o n d i t i o n s C ( x , i ) (or a n o t h e r c o n d i t i o n like the f l u x c o n d i t i o n dC(\,t)/dx . . .
w h e r e χ e 9 Ω ) . T h e i m p o r t a n t f a c t is t h a t t h e s e c o n d i t i o n s d o not c h a n g e t h e p h y s i c s o f t h e d i f f u s i o n but o n l y g i v e a p a r t i c u l a r c o n f i g u r a t i o n to t h e s y s t e m . A s t h e r e can b e an infinite n u m b e r o f initial or limit c o n d i t i o n s , the n u m b e r o f s o l u t i o n s f o r t h e d i f f e r e n t i a l s y s t e m b e c o m e s i n f i n i t e a n d g e n e r a l l y c a n n o t b e s o l v e d analytically. All s o l u t i o n s can b e s e e n a s d e n s i t y f u n c t i o n s b e c a u s e c o n c e n t r a t i o n in s p a c e dx can be seen as a p r o b a b i l i t y t o f i n d p a r t i c l e s in dx. In this part, w e shall p r o v e that all t i m e - d i f f u s i o n s y s t e m s can be seen as t h e s u m o f e l e m e n t a r y s a n d w i c h p r o c e s s e s c e n t e r e d on e a c h p a r t o f a grid e q u a l l y s p a c e d .
5.1 Relation between stochastic microscopic diffusion state and microscopic deterministic diffusion state Let us c o n s i d e r the d i f f u s i v e s y s t e m as a m i c r o s c o p i c o n e and n o t e η t h e n u m b e r of particles at t i m e t at p o s i t i o n x,(t), / e { l . . . « } . A s the position c a n n o t b e k n o w n e x a c t l y ( H e i s e n b e r g ' s p r i n c i p l e ) , w e can c o n s i d e r that the p r o b a b i l i t y to f i n d a p a r t i c l e is g i v e n by: ( 1 4 ) <p(x,,t;h) = — e x p hyl 2π
w h e r e h represents the lack of precision in the
x,(t) determination (if h-> 0 , then
φ(χ,, f; h) δχ ( 0 , a n d t h e p o s i t i o n is p e r f e c t l y k n o w n ) . W e shall n o w link t h e m i c r o s c o p i c i n t e r p r e t a t i o n t o t h e m a c r o s c o p i c s y s t e m . Let u s n o t e C(x,t) t h e s o l u t i o n o f t h e P D E p r o b l e m a n d , n ) t h e s o l u t i o n g i v e n b y t h e m i c r o s c o p i c o n e . A s a c o n s e q u e n c e , u s i n g t h e G a u s s i a n K e r n e l D e n s i t y E s t i m a t e w e get: C(x,t,n-,h) = - - Y ( p ( x . , f , h ) . n h t l ( 1 5 ) T h e c o n s t a n t h is c h o s e n t o f i t w e l l w i t h t h e m a c r o s c o p i c c o n c e n t r a t i o n t h a t can b e e x p l a i n e d b y a L2 n o r m m i n i m i z a t i o n : m i n | φ , ί , « ; / ; ) - φ , / ) | \ helR It can b e s h o w n t h a t : l i m m i n l lc( * ' ' > '7;/ ?) -cM ( 1 6 ) = 0 f o r h - > 0 . ( 1 7 ) m e a n i n g that a s t o c h a s t i c m i c r o s c o p i c s y s t e m t e n d s t o w a r d s a m a c r o s c o p i c d e t e r m i n i s t i c o n e w h e n t h e n u m b e r o f p a r t i c l e s t e n d s t o w a r d s infinity.
5.2. Decomposition of complex macroscopic diffusion systems into elementary mesoscopic diffusion microstates. Let us n o w s u p p o s e t h a t t h e c o n c e n t r a t i o n C(x,t) is k n o w n ( f o r e x a m p l e Fig. 3 ( b i ) o r Fig. 3(b5). W e d i s c r e t i s e s p a c e in i n t e r v a l s of length Δχ . O n e a c h interval l a b e l e d by η s u f f i x , t h e c o n c e n t r a t i o n is a c o n s t a n t m o d e l e d by a r e c t a n g u l a r f u n c t i o n H(XN,AX) ( s e e Fig. 3 ( a l ) ) so that the c o n c e n t r a t i o n €„(χ,ΐ,Δχ) on t h e central interval xn is g i v e n by ( f o r e x a m p l e Fig. 3 ( a l ) or Fig. 3 ( a 5 ) ) : {x% U Δχ) = C(xn, t )H(,yh , Δ.ν). ( 1 8 ) T h e n t h e d i s c r e t i s e d c o n c e n t r a t i o n c(x,/,Av) is given by: σ ( χ , ί , Δ χ ) = Σσ. (χ> ' >Α χ) · (1 9) I ' l l O n e a c h interval, t h e d i f f u s i o n g o v e r n e d by E q . 2 still o c c u r s , but w i t h t h e initial c o n d i t i o n on
128 Μ. Bigereile and A. lost /Int.J. Nonl. Sei. Num. Simulation 5(2), 121-134, 2004
(20) these intervals given by:
Γ Cn{x,t,Ax) = C(x„,t,^) {C„{x,t>Ax)=0
Vx e [x„ - A x / 2 , x „ + Δχ / 2 J Vx 0 - Ax/2,x„ + Ax! 2~\
When interval / is discretised in the time interval [/,/ + Δ/] , labeled by m suffix ,]), we obtain: iCnm{x,tm,Ax) = C{xn>tm,Ax) {C„Jx,tm,Ax)= 0 Vx e [x - Ax / 2, χ + Ax 12\ r η w Λ ο A · ( 2 1 ) Vx g [χ„ - Ax/2,χ„ + Ax/ 2J
This means that the initial PDE problem can be modeled by a great number of cells on which the initial concentration is constant where Eq.2 holds.
The solution to Eq.2 on this "PDE-cell" is then given by:
Cmm(x,t,A= e x p f - i ^ V
(22) After integration of Eq.22, the final solution is:
Ax / 2 - (x - xn) erj •
C„Jx,t,Ax) = ~C{x„J„,,Ax)
dx
with erf{x) , the error function given by 2 « / ( * ) = —,= |exp(- u2) du . •ν TT 0 Theorem lim I —>x Μ λ ) 4 Dt (24) This means that for a long enough diffusion time, the PDE-cell is physically equivalent to the well-known problem of sandwich diffusion centered on the x„ value. The convergence rate of theorem 1 will be analyzed in the next chapter.
Theorem 2: The solutions C(x,t) to all the
A * 8 -10 -5 o.,! t=1.5 OK ; t=i .9' 0 s 10 •10 -S 0 S 10 •10 -5 0 S 10 •10 -5 0 S 10 -10 -5 0 5 10 8 « 650.3 -10 - 5 0 5 10
Fig. 3. Evaluation of a diffusion problem for a thin material diffusing in a random medium. Values of C(xl,,tm,Ax) for different diffusion times are plotted on Fig. (a/) to Fig. (an). Fig. (a,) (t = t0= 0) and Fig.(as) (t = t] = l J are calculated by integration oj theoretical values Clh(x,0)=S0 (Fig. (aJ and
Clh(x,l) (Fig.(as)). Fig. (bz), (b3), (b4), (b6), (by), (ba) represent the sum of C(xn,tm, Ax), i.e. Cm(x,t), given on the left figure (circle). The theoretical values given by the PDE problems are plotted on filled line.
I S S N 1 5 6 5 - 1 3 3 9 I n t e r n a t i o n a l J o u r n a l o f N o n l i n e a r S c i e n c e s and N u m e r i c a l S i m u l a t i o n 5 ( 2 ) , 1 2 1 - 1 3 4 , 2 0 0 4 129 d i f f u s i o n p r o b l e m s in i n f i n i t e m e d i a (or w h e r e c o n c e n t r a t i o n in all t h e m e d i a d o e s not c h a n g e ) w h i c h o b e y E q . 2 , are s o l v e d by E q . 2 3 on e a c h P D E - c e l l a n d t h e n t h e m a c r o s c o p i c s o l u t i o n is given by: c M = l i m Y φ „ , ? 0, Δ χ )
V4πϋϊ .
J
exp(* -
Xn)
2 4 Dtdx
( 2 5 ) w h e r e C ( x „ , t0, Δχ) is t h e initial c o n c e n t r a t i o n in each cell. T o illustrate this t h e o r e m t h a t is t h e c e n t r a l point of o u r p h y s i c a l a p p r o a c h t o stability, w e shall treat t h e m a c r o s c o p i c p r o b l e m o f initial thin film t h a t d i f f u s e s in a n i n f i n i t e m a t r i x . E q . 3 g i v e s the s o l u t i o n t o t h e m a c r o s c o p i c d i f f u s i o n p r o b l e m . T h e m e t h o d d e s c r i b e d b e l o w is then applied with t h e f o l l o w i n g p a r a m e t e r s : D = 0.01 , Δχ = 1 , I e [0...2] , tm e {θ,ΐ} . » = { - 1 0 , - 9 , ...,9,10}. At each t i m e w h e r e t = tm , t h e v a l u e s o f C(xn,tm,Ax) h a v e t o b e n u m e r i c a l l y a s s e s s e d . W e c a l c u l a t e t h e m e a n o f the c o n c e n t r a t i o nC(xll,tm,Ax) on the interval
[x„ - A x / 2 , xn + Δ χ / 2 ] g i v e n by:
C(xn,tm,Ax) =
Ax/2
tsx^nDt„,
I exp(
x~
x„ f dx.
(26)
and then f r o m E q . l 8C(x„,tm
, Δχ) = C(x„,t„,,Ax)H{x„,Ax) (for t
m = 0 ,C{x0j) = Su
, Fig 3 (b|) and
C{x„,t0,Ax)=H(xN,Ax)/Ax, Fig. 3 ( a , ) ) . T h e n
Eq.23 is a p p l i e d to c a l c u l a t e t h e c o n c e n t r a t i o n on each cell at d i f f e r e n t t i m e s Cnm(x,t,Ax). Fig.
3.(a), r e p r e s e n t s t h e d i f f e r e n t v a l u e s o f C,,„,(*,/,Δχ) f o r m = 0 ( F i g . 3 (a,), t o F i g . 3 (a4)) and m = 1 ( F i g . 3 ( a5) to Fig.3 (a8)). T o v e r i f y the c o r r e s p o n d e n c e w i t h t h e t h e o r e t i c a l v a l u e (3), w e c a l c u l a t e t h e c o n c e n t r a t i o n o b t a i n e d by o u r " M e s o s c o p i c " m o d e l f r o m t h e o r e m 2:
^>)=ςΨφ4 Τ-ρ
4D(t Oy
( 2 7 )dx
and t h e n c o m p a r e w i t h t h e t h e o r e t i c a l v a l u e s C,h(x,t) g i v e n by E q . 3 . F i g . 3 ( b , ) t o Fig.3 (b8)represent Cm(x,t) and C,h(x,t) for different
d i f f u s i o n t i m e s . A s c a n be o b s e r v e d , t h e m e s o s c o p i c p r o b l e m c o n v e r g e s t o w a r d s t h e analytical f o r m u l a f o r a s u f f i c i e n t d i f f u s i o n t i m e ( n o t e that t h e " t h r e s h o l d " t i m e d e p e n d s on Δ χ a n d the d i f f u s i o n m e d i u m p r o p e r t i e s : t h e l o w e r Δχ or t h e h i g h e r D , t h e l o w e r t h i s t h r e s h o l d , cf. t h e o r e m 2). W e h a v e t h e n p r o v e d that a m a c r o s c o p i c c o m p l e x d i f f u s i o n p r o b l e m can b e d e c o m p o s e d into a set o f e l e m e n t a r y d i f f u s i o n p r o c e s s e s c o n t r o l l e d o n an e q u a l l y s p a c e d grid w h i c h leads t o a G a u s s i a n p r o b a b i l i t y D e n s i t y F u n c t i o n . T h i s d e c o m p o s i t i o n illustrates M . S . El N a s c h i e ' s idea t h a t e l e m e n t a r y p r o c e s s e s o f m o r e c o m p l e x s y s t e m s a r e G a u s s i a n [11 ] .5.3. 5.3.
Fig. 4. Three adjacent cells xl,x2,x1 of the space
discretised grid Δχ = 1 on which mesoscopic
simulations are processed until the instability
criterion is reached. Gaussian curves Gl, G2 and
G} with standard deviation Ax = 42Dt =1
(D = 1, / = 1 /2 , i.e. λ = \ 12) are centred on each
cell.
Analysis of an elementary mesoscopic diffusion micro-state Let us n o w c o n s i d e r t h r e e a d j a c e n t cells x , , x2, x3 ( F i g . 4 ) . W i t h o u t lack o f g e n e r a l i t y , w e shall c o n s i d e r that
Φι. L > Δχ) = C(x
2, t
m,
Ax) = c(x3,t
m, Ax) a n d
tm = 0 . S u p p o s i n g a l o n g e n o u g h d i f f u s i o n t i m e ,w e get t h r e e G a u s s i a n c u r v e s w i t h the s a m e area, c e n t e r e d on xl,x2,xJ v a l u e s and w i t h a
Λ/2 Dt s t a n d a r d d e v i a t i o n . T o o b t a i n e x a c t l y t h e s t a b i l i t y - i n s t a b i l i t y t h r e s h o l d , f r o m E q . 1 3 w e
130 Μ. Bigerelle and A. lost /Int.J. Nonl. Sei. N u m . Simulation 5(2), 121-134, 2 0 0 4
must have Ψ(Ρ) = 1 which involves Δχ = <JlDt (if 4lDt > Δχ then the scheme is unstable). Fig.5 represents the three Gaussian curves at the instability threshold. Let us now analyze very precisely this critical density function. Each elementary cell d i f f u s e s on the adjacent intervals (a probability calculus shows that 24 % of the mass transfer is on adjacent cells). Let us now consider the x, interval. Concentration is the summation of all the Gaussian curves on these intervals. It is important to notice that the concentration in cell x2 only results from the
flux from cell x, and cell x3 . An important
property is that, at the instability threshold, the inflexion points of the Gaussian curves G, and
G3 lie in the center of the interval x2.
The following mathematical properties can be stated: d2C] 0(x, Δχ2 /2D,Ax)/dx2 < 0 d2C3 0(x, Δχ2 /2D,Ax)/ dx2 > 0 , (28) Vx e [x2 - Δχ / 2, x2 ] d2Ct0 (x2, Δχ2 / 2D, Δχ)/ dx2 = 0 d2C10(x2,Ax2 12 D,Ax)/dx2 =θ' a2C] 0( x , A x2 l2D,Ax]ldx2 > 0 d2C30(x, Ax2 /2D,Δχ)/ 3x2 < 0 . (30) Vx e [x2, x2 + Ax / 2]
From Eq.2, d2C: j (x, t, Δχ)/ dx2 oc dCij(x,t,Ax)ldt
then 50 % of the dx micro intervals ( d x « Δχ ) of x2 interval get a positive temporal gradient
( dCi0(x,t,Ax)/dt > 0 and dC}0(x,t,Ax)/dt > 0 ),
and 50% a negative gradient
( dC]0(x,t,Ax)/dt < 0 and C30(x,t,Ax)/dt < 0 ).
After the stability threshold, more than 50% of micro cells get a negative gradient. This clearly means that the number of particles issued from adjacent cells x, and x3 decreases with time
(and so does the configuration entropy of the system x2) , which does not comply with the
second principle of t h e r m o d y n a m i c s for a diffusive s y s t e m . . . T h e mathematical criterion of stability for the explicit scheme is then physically related to the production of entropy: if the scheme is unstable then the production of
(29)
entropy on each discretised cell is negative
( dSlim(x,t,Ax)/dt < 0 ). In chapter 7, we shall
process a M o n t e Carlo simulation of diffusion to illustrate the mathematical tools of stability.
6. Numerical instability: an infringement
of the second principle of
thermodynamics.
In a monodimensional case, a P D E cell gets t w o adjacent P D E cells that will d i f f u s e on this cell (see Fig.4).
We shall consider particles issued f r o m the adjacent cells. At time origin, it w a s shown in the previous paragraph that no particle from the adjacent cells is present in the P D E cell. During the diffusion process, these particles will d i f f u s e on the cell with respect to time. Let us now formulate the configuration of cells xl or x3 in Fig.4. At the origin, entropy equals zero because no particle from the a d j a c e n t cells is present. Then entropy will increase thanks to diffusion. W e shall now introduce the mathematical formalism of entropy evaluation. Based on the statistical thermodynamics, the P D E cell will be divided into k micro-systems of length δχ ,
{Ax = köx,h>> 1) we call the ' S u b - P D E cells'.
On each S u b - P D E cell, the probability of finding particles f r o m the adjacent cells is evaluated and is of course time dependent. If the concentration in the P D E cell obeys the Gaussian hypothesis, the probability of having particles in an interval of length δχ, (<£χ«Δχ) located on χ in the PDE-cell ( x e [ - Δ χ / 2 . . . Δ χ / 2 ] ) is given by: P(x,t , Αχ, δχ) = erf\ Δχ - χ + δχ •jÄDt \ -erf ^ Δ χ - χ ^ Λ/4Έ .(31) Then the entropy configuration on each Sub PDE-cell is given by:
AS{x,t,Ax^x)=-R P(x,t,Ax,5x)logP(x,t,Ax,&),
(32) where R is the gas constant.
By means of entropy additivity, the entropy 011 any PDE-cell is then given by:
I S S N 1 5 6 5 - 1 3 3 9 I n t e r n a t i o n a l J o u r n a l o f N o n l i n e a r S c i e n c e s a n d N u m e r i c a l S i m u l a t i o n 5 ( 2 ) , 1 2 1 - 1 3 4 , 2 0 0 4 131 Σ
kl 2
Σ
i / 2 + Ierf
χ log Δχ 1 -/ - 1 λ/407 - erff ,
- i - 1Ax 1
- i - 1 Ik)
λ/407 er/ Δχ 1 -Λ/407 - e r / Δχ 1-k
yfÄDl
( 3 3 ) and finally, t h e cell e n t r o p y is o b t a i n e d w h e nδχ - > 0 m e a n i n g that A -> oo : Δ5>(/,Δχ) = lim AS(t,Ax,k) (34) 1 0.8 >, 0.6 Q. Ο Έ 0.4 UJ 0.2 0 0 9 0 9 0 8 0 7 0 6 0 0 8 0 7 0 6 0 0 8 0 7 0 6 0
/
0 8 0 7 0 6 0 1 0 3 0 5 0 7 0 9 2 4 6 Equilibrium criterion 10Fig. 5. Entropy calculation for the cell x2
shown in Fig. 4 by numerical estimation oj
Eq.33 for a configuration
k = 30, D = 1, Δχ = 1,
/ e {0,0.001,0.002,•••,9.999,10} versus the
values of stability criterion At/Ax2. Entropy
increases for the value of At / Ax2
= 0.5
(graph zoom), and then entropy decreases.
T h e o r e m 3:dAS(t, Ax)/dt >0 i f < Ax ,t > 0 , (35)
dAS(t,Ax)/dt = 0 if flDt = Ax , (36)
dAS(t, Ax)/dt <0 if flDt > Ax . (37)
In instability c a s e s , e n t r o p y will d e c r e a s e in t i m e w h i c h p h y s i c a l l y c o n t r a d i c t s t h e s e c o n d principle of t h e r m o d y n a m i c s ( T a b l e 2). W e shall n o w e s t i m a t e the e n t r o p y o f t h e cell x2 s h o w n in Fig.5 by the n u m e r i c a l e s t i m a t i o n of Kq. 33 for a particular c o n f i g u r a t i o n , k = 3 0 , D= I , Λχ - I, te {θ,0.001,0.002,· • ·,9.999,10}.{INSTABILITY
yflDt
STABILITY— - > Δ Χ Scale Anti Entropy Process
T a b l e 2 Entropy Production Fig.6 r e p r e s e n t s t h e v a r i a t i o n o f t h e e n t r o p y o f cell x2 v e r s u s t h e v a l u e s o f t h e stability c r i t e r i o n D At / Ax2. A s c a n b e o b s e r v e d , e n t r o p y i n c r e a s e s u p t o At/Ax2 = 0 . 5 . U n d e r this v a l u e t h e n u m e r i c a l s c h e m e is s t a b l e , w h i l e o v e r it t h e n u m e r i c a l s c h e m e is u n s t a b l e and e n t r o p y d e c r e a s e s . T h i s is an illustration o f the P r i g o g i n e and B r u s s e l s s c h o o l w h i c h s h o w s via stability
Fig. 6. Diffusion resulting from the three
adjacent Cell-PDE modelled by a Brownian
motion compared with the analytical solution
under Gaussian hypothesis. n
sis the number oj
jumps of the particles, which corresponds to the
time leading to the scheme instability (cf. Fig. 5).
a n a l y s i s t h a t n o n - l i n e a r b i f u r c a t i o n can a b r u p t l y d e c r e a s e local e n t r o p y [12]. In f a c t , f o r a g i v e n o b s e r v a t i o n t i m e At, if Δχ d e c r e a s e s the p r o b a b i l i t y o f f i n d i n g p a r t i c l e s on Ax t e n d s to d e c r e a s e a n d t h e n w e t e n d t o a t i m e reversal t h a t v i o l a t e s the t i m e s y m m e t r y b r e a k i n g i n d u c e d by t h e s e c o n d l a w o f t h e r m o d y n a m i c s [13].7. Monte Carlo simulation
T h e B r o w n i a n m o t i o n can also m o d e l t h e solution r e s u l t i n g f r o m the P D E - c e l l . P h y s i c a l l y s p e a k i n g , ρ p a r t i c l e s are u n i f o r m l y r a n d o m l y d i s t r i b u t e d on t h e interval [x„ - Δ χ 7 2 , χ „ +Ax/2], d i v i d e d into k s u b d i v i s i o n s o f length a at t i m e
132 Μ . B i g e r e i l e a n d A. l o s t /Int.J. N o n l . Sei. N u m . S i m u l a t i o n 5 ( 2 ) , 1 2 1 - 1 3 4 , 2 0 0 4
tn (Ax = ka,k» 1). Then for each ρ particle,
ns jumps are processed on the left or on the
right with a probability of one half. From the microscopic theory of diffusion, the diffusion coefficient is:
D = a2 Γ / 2 , (38)
where Γ is the j u m p frequency of a particle from a micro-cell of length a to an adjacent micro cell.
The stability condition can then be expressed by including (10) into (38): Δ/ 1 n,=l n=l^ JOOOi 25003CXXX < ι2Γ (39) 1, £ = 1 0 0 , a = 10" (Ax)2 Numerical application: D p = 100000.
Then we get: Δχ = 1, ns = ΓΔί (where ns is the
total number of jumps during time At) and the stability criterion (sc) becomes
ns <(Δχ/α)2 = nf =10000 . Firstly, we shall
verify the convergence rate of theorem 1. We shall study the diffusion time effect between three adjacent cells for the values of
ns € {l, 10,100,1000, lOOOO} . Fig.7 shows the
concentration curves for these different evolution times and the theoretical Gaussian curves given by Eq.24 plotted for each concentration map. As can be observed, the convergence rate to the Gaussian curve is well adapted if ns >1000. As our theory induces the
Gaussian approximation PDF for the case - n" = 10000, then the Gaussian PDF is an adequate assumption.
We shall now study the case ns = nf for
k = 1000, then nf =1000000 more particularly. This simulation corresponds to a fine resolution of the diffusion problem on three adjacent cells at the instability threshold. Fig. 7 represents the theoretical solution (filled lines) under the Gaussian assumption (Eq.24) and the results from the Monte Carlo simulations. As can be observed, the Monte Carlo results converge towards the solution of the Gaussian approximation given by Eq.24 replacing the more complex Eq.23. This means that the Gaussian convergence claimed in theorem 1 is well accepted by our stability criterion and then
n,=l <y 300» C Δ η -inΜ ·%"~ :50£X ,υ 2000C E9 1S00C Bffl 1000C HftS. 500C A n,=io iE rxjj i5ooc lift 10001 n,= lO i X> 30CKX C Α 1ΛΛ 250UC ι Μ Oj-ιυυ 2oooc m '50tx c Jrh ιοοα MA sooc Δ iu=100 2500C « "S , w u 2000 ΜΛ 15001 m 100CK η,= 100 Δ id . I St Κ 2000C 1 113=1000 1500C \ 1000C ft 5000 A 2000C Ä 1^=1000 1500C WH 1000C Jaffa 5000 n,=1000 ,
1
\ iV=10000 a 5000A
^=loooo 5000 n,=1000^ - 6 - 3 - 1 0 2 - 4 - 2 - 0 1 ! - 5 - 3 - 1 0 2 4 - 4 - 2 - 0 1 3 -S -3 -1 0 - 4 - 2 - 0 1Fig. 7. Comparison of the solution of the diffusion on three adjacent Cell-PDE of length Δχ = 1000 a modelled by a Brownian motion with the analytical solution under Gaussian hypotheses. The number of jumps nf = 1000000 of length a corresponds to the time leading to the scheme instability.
all our hypotheses are well posed. As a consequence, all our conclusions about stability and the relations with a physical interpretation speak for themselves in the practical case of finite simulation.
8. Conclusion
In this paper, we have shown that there exists a relation between the stability criterion that guarantees the convergence of the partial derivative equation to the solution and the physical interpretation at a mesoscopic scale under the size of discretisation. This interpretation is related to an infringement of the fundamental second principle of thermodynamics. Regarding the high number of stability studies found in the mathematical and physical bibliography, the stability threshold allows us to give a space and time definition onto which scales are well adapted to treat the physical problems at a mesoscopic scale. These criteria are pertinent to choose discretised time and space during a Monte Carlo simulation. The stability criterion gives a numerical evaluation of the threshold between mesoscopic and microscopic simulations. To build this theory,
I S S N 1 5 6 5 - 1 3 3 9 I n t e r n a t i o n a l J o u r n a l o f N o n l i n e a r S c i e n c e s a n d N u m e r i c a l S i m u l a t i o n 5 ( 2 ) , 1 2 1 - 1 3 4 , 2 0 0 4 133
an original mathematical tool was constructed: we have shown that complex macroscopic problems can be treated as sets of mesoscopic problems that can be solved analytically. This approach consists in choosing an adequate model for the physical phenomenon on the elementary cells to solve the physical problem in its integrity. We think that each element of discretisation must have some physical properties rather than mathematical asymptotic properties to solve physical differential problems. On the other hand it is in total agreement with Hadamard Conference (Zürich,
1917 cited in [14]) which disagrees with the common idea that a non-analytical partial derivative equation can be replaced by an analytical function without any significant effect, proving that an approximation could significantly disturb the solution.
In our case we have studied a linear equation that in fact could not lead to chaos. However we have shown that its discretisation scheme could lead to a chaotic (unstable) solution if discretization do not respect some physical considerations. This clearly means that diffusion equation, which leads to the differential equation, has lost a part of chaos due to asymptotic mathematical considerations. On the contrary, the diffusion map that we have proved to be physically realistic is highly non linear. From the information contained in these non linearity's, a space-time relation was stated proving that non linearity allows us to explore the deep properties of diffusion : "If everything
were linear, nothing would influence nothing",
Einstein to Schrödinger (Cited in [13] with Author's comments about non linearity's effects in the space-time description).
We have only dealt with diffusion in a one-dimension space. We have studied the bi and three-dimensional cases and all results are confirmed. By applying our space-compactification principle [15, 16], we have found the same results using an isomorphism between concentration and lnformions. These results, in agreement with Penrose's views on the computability of complex system [17-18], will be tackled in another paper.
A c k n o w l e d g e m e n t s
We wish to thank Veronique Hague for her assistance in English and Professor El Naschie for his stimulating remarks.
R e f e r e n c e s
1. Dautray R, Lions JL. Analyse mathematique
et Calcul numerique. Modele physique, Vol. 1.
Paris:Masson;1987.
2. Dautray R, Lions JL. Analyse mathematique
et Calcul numerique; Evolution, Fourrier, Laplace, Vol.7. Paris:Masson;1992.
3. Dautray R, Lions JL. Analyse mathematique
et Calcul numerique; Numerique, transport
Vol.9. Paris:Masson; 1992.
4. Forsythe GE, Wasow WR. Finite difference
methods for partial differential equations.
New York: Wiley & sons; 1960.
5. Peitgen HO, Jürgens Η, Saupe D. Chaos and
Fractals New Frontiers of Science. Berlin:
Springer Verlag; 1992.
6. Adda Y, Philibert J. La diffusion dans les
solides, Tome II. Paris:Bibliotheque des
Sciences et Techniques Nucleaires; 1966. 7. Einstein A. Investigation on the Theory of
Brownian Motion. New-York: Dover Pub.;
1956.
8. Bruhat G. Thermodynamique. Paris: Masson; 1962.
9. El Naschie M. S. Remarks on superstrings, fractal gravity, Wagasawa's diffusion and Cantorian Spacetime. Chaos Solitons &
Fractals 1997;8:1873-86.
10.Stewart I. Does God play dice? The new
Mathematics of Chaos. London: Penguin
Books; 1997
1 l.EI Naschie Μ S. Branching polymer and the fractal Cantorian Space. Chaos Solitons &
Fractals 1998;9:135-41.
12.Prigogine I. From being to becoming. New-York: Freeman; 1980.
13.El Naschie Μ S. Time symmetry breaking, duality and Cantorian space-time. Chaos
Solitons & Fractals 1996;7;4:499-518.
14.Gray J. Le Defi de Hilbert. France: Dunod; 2003.
134 Μ. Bigereile and A. l o s t / I n t . J . Non!. Sei. N u m . Simulation 5(2), 121-134, 2 0 0 4
