finite difference method spatial and time discretization...

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Solutions to the diffusion equation

Numerical integration (not tested)

● finite difference method

● spatial and time discretization

● initial and boundary conditions

● stability

Analytical solution for special cases

● plane source

● thin film on a semi-infinite substrate

● diffusion pair

● constant surface composition


Numerical integration of the diffusion equation (I)Finite difference method. Spatial Discretization. Internal nodes.

RiiLiii

ti

tti AJAJV

t

CC11

Δ

Δ

For 1D thermal conduction let’s “discretize” the 1D spatialdomain into N small finite spans, i = 1, …, N:

Balance of particles for an internal (i = 2 N-1) volume Vi

(change in concentration during t due to the particle exchangewith adjacent cells):

x

Vi

Jii+1

ARAL

i-1 i i+1Ji-1i

x

CCDJ

ti

ti

ii Δ1

1

x

CCDJ

ti

ti

ii Δ1

1

R

ti

ti

L

ti

ti

i

ti

Δtti A

x

CCDA

x

CCDV

t

CC

ΔΔΔ11

Vi = xyz, AL=AR=yz

First Fick’s law:

211

Δ

Δ

2

Δ x

CCCD

t

CC ti

ti

ti

ti

tti


Numerical integration of the diffusion equation (II)Finite difference method. Spatial Discretization. Internal nodes.

211

Δ

2

Δ x

CCCD

t

CC ti

ti

ti

ti

Δtti

211Δ

Δ

2Δ

x

CCCDtCC

ti

ti

tit

itt

i

Using this equation we can calculate unknown concentration attime t+t if we know concentrations at previous timestep t.

The finite difference scheme that we discussed is called theexplicit Forward Time Centered Space (FTCS) method, sincethe unknown temperature at node i is given explicitly in terms ofonly known temperatures computed at the previous timestep.

x or i

tor

n

known points

new point

There is pair by pair cancellation of terms in the FTCS equation thatensures that whatever the number of diffusing species leaves onevolume is picked up by its next-door neighbor.


Numerical integration of the diffusion equation (III)Finite difference method. Spatial Discretization. End nodes.

Let’s consider the first node, i = 1 and let’s place the nodeat the surface.

x

Vi

J23

AR

Node 1 Node 2 Node 3

J12

Balance of particles for volume V1 that is represented bythe end node:

R

ttt

AJVt

CC211

1Δ

1

Δ

R

tt

Ax

CCDJ

Δ12

21

R

ttttt

Ax

CCDV

t

CC

ΔΔ12

11

Δ1

First Fick’s law:

V1 = VN = Vi = 2N-1/2, xAR = xyz = Vi = 2N-1 = 2V1

2121

Δ1

Δ2

Δ x

CCD

t

CC ttttt


Numerical integration of the diffusion equation (IV)Finite difference method. Spatial Discretization. End nodes.

212

1Δ

1 Δ2Δ

x

CCDtCC

ttttt

Similarly, for another end node N, one can derive:

2121

Δ1

Δ2

Δ x

CCD

t

CC ttttt

21Δ

Δ2Δ

x

CCDtCC

tN-

tNt

Ntt

N

If an initial concentration profile at time t = 0 is given, wecan use the derived equations for all nodes, from 1 to N, tocalculate concentration profile at time t, 2t, 3t, etc. upto the time that we are interested in.


Stability criterion

For a position-dependent diffusion coefficient, it should beevaluated at the interfaces between the volumes Vi in orderto preserve the conservation of the total number of particles

e.g., for sample with non-uniform T distribution

ti

tii TDTDD 12/1 2

1

211Δ

Δ

2Δ

x

CCCDtCC

ti

ti

tit

itt

i

2

121121Δ

ΔΔ x

CCDCCD

t

CC ti

ti/i

ti

ti/i

ti

tti

Position (C, T,…) dependent D(x,t)

Assuming that the coefficients of the differential equationare constant (or so slowly varying as to be consideredconstant) the stability criterion for

is 1Δ

Δ22

x

tD

Physical interpretation: the maximum timestep is, up to anumerical factor, the diffusion time across a cell of widthx (remember Einstein relation D = s2/2dt).

This condition is referred to as the VonNeumann stability condition.


0

Lxx

C

No mass transfer through the boundary:

Constant concentration at the boundary CL = Cf and we canuse the following boundary condition:

fLxCC

Simple boundary conditions

N-2 N-1 N N+1

JNN+1= 0

fictive node

CN CN+1 = CN


Diffusion equation in 2D

2

2

2

2

y

x,y,tCD

x

x,y,tCD

t

x,y,tCyx

Boundary condition: x,y,tCx,y,tC bboundary

x,yCx,y,C init0Initial condition:

x,y,tCb

i1i 1i

1j

j

1jyΔ

xΔ

i = 1 i = Nx

j = 1

j = Ny


Diffusion equation in 2D: explicit FTCS scheme

2

2

2

2

y

x,y,tCD

x

x,y,tCD

t

x,y,tCyx

t

,jiti,j

t,ji

x

ti,j

tti,j CCC

x

D

t

CC112

Δ

2ΔΔ

t

,jiti,j

t,ji

xti,j

tti,j CCC

x

tDCC 112

Δ 2Δ

Δ

2

1

Δ

Δ

Δ

Δ22

y

tD

x

tD yxStability:

tD~r D Δ4Δ 2

ti,j

ti,j

ti,j

y CCCy

D112

2Δ

ti,j

ti,j

ti,j

y CCCy

tD112

2Δ

Δ


Diffusion equation in 3D: explicit FTCS scheme

t

,j,kiti,j,k

t,j,ki

x

ti,j,k

tti,j,k CCC

x

D

t

CC112

Δ

2ΔΔ

2

2

x

x,y,z,tCD

t

x,y,z,tCx

t

,j,kiti,j,k

t,j,ki

xti,j,k

tti,j,k CCC

x

tDCC 112

Δ 2Δ

Δ

2

1

Δ

Δ

Δ

Δ

Δ

Δ222

z

tD

y

tD

x

tD zyxStability:

tDr D 6~3

2

2

2

2

z

x,y,z,tCD

y

x,y,z,tCD zy

ti,j,k-

ti,j,k

ti,j,k

zt,ki,j

ti,j,k

t,ki,j

y CCCz

DCCC

y

D112112

2Δ

2Δ

ti,j,k-

ti,j,k

ti,j,k

zt,ki,j

ti,j,k

t,ki,j

y CCCz

tDCCC

y

tD112112

2Δ

Δ2

Δ

Δ


Possible algorithm for solving 1D diffusion equation

1. Initialize diffusion constant D, system size L, step of spatialdiscretization x, time-step h, and the total time of the simulation.You can use N ~ 100 steps in spatial discretization. Make sure thatstability criterion is satisfied.

3. At each node position except the two surface nodes calculate newconcentration using the finite difference method described in thelecture notes, use a dummy array to store new concentration.

2. Define an initial concentration profile, Ci0. Use an array to represent

the concentration at discrete points x0, x1, …, xN and set the value ateach point equal to the initial concentration at that point.

5. Copy the dummy array into your concentration array.

211 2

x

CCCDtCC

ti

ti

tit

itt

i

for i = 2, …, N-1

4. Calculate concentration for the two surface nodes using boundaryconditions appropriate for the problem of your choice. For example,for zero-flux surfaces we have:

212

11 2x

CCDtCC

ttttt

212

x

CCDtCC

tN

tNt

Ntt

N

Note, that 2 in the right term of these formulas result from the factthat the thickness of the first and last slices is x/2 rather than x.

7. If the time is less than the time of the simulation, go to 3 and repeat.

6. Update current time = time + h

8. Write data for plotting final concentration profile versus position.You can also write data several time during the simulation if you wantto see the evolution of the concentration profile with time.


While the numerical method described above can be applied tosolve the diffusion equation for any initial and boundaryconditions, in many special cases it is possible to derive analyticsolutions as well. The advantage of analytical solution – does notinvolve long computations, gives clear picture of the dependenceof the solution on different parameters in the whole range of xand t.

Analytical Solutions of the Diffusion Equation

2

2

x

tx,CD

t

tx,C

Let’s consider the simple case of solute atoms initiallyinserted into the middle of an infinite one-dimensionalsystem (plane source).

All solute is in one plane initially

With time the solute atoms will diffuse from the plane.The redistribution of the atoms can be described by thefollowing solution of the diffusion equation:

4Dt

xexp

t

Atx,C

2


Analytical solution for plane source

2

2

x

tx,CD

t

tx,C

We can show by differentiation that for the plane source inan infinite system

4Dt

xexp

t

Atx,C

2

is the solution of thediffusion equation


Analytical Solutions of the Diffusion Equation

4Dt

xexp

t

Atx,C

2


Analytical solution for plane source

Coefficient A we can find from the condition that the totalquantity of solute per unit of the cross-section area S is

dx4Dt

xexp

t

Adxtx,CS

M2

Using substitution of variable4Dt

xy

π4DASdyyexpt

4DtASM 2

dy4Dtdx

D 4 πS

MA

4Dt

xexp

Dt π2 S

Mtx,C

2

The solution is a Gaussian concentration profile whose width increases with time

t1

t2 > t1


Side-note: Derivation of Einstein relation for 1D case (not tested)

Diffusion equation:

Let’s multiply the diffusion equation by x2 and integrate over space:

2

2

x

t,xCD

t

t,xC

dx

x

t,xCxDdxt,xCx

t 2

222

tx 2

dx

x

t,xCx

xDdx

x

t,xCxD

t

tx2

2

22

2

dxt,xCx

xD2dxt,xxC

xD20

dxx

t,xCxD2

x

t,xCDxdx

x

t,xC

x

xD 2

2

D2dxt,xCD2t,xDxC2

ADt2tx 2 for 1D diffusion

Let’s consider diffusion of particles that are initially concentrated at the

origin of our coordination frame, - Dirac delta function x0,xC

Solution:

4Dt

xexp

πDt2

1tx,C

2


Side-note: Derivation of Einstein relation for d dimensions (not tested)

Diffusion equation:

Let’s multiply the diffusion equation by r2 and integrate over space:

t,rCDt

t,rC 2

rdt,rCrDrdt,rCrt

222

tr 2

rdt,rCrDrdt,rCrD

t

tr222

2

rdt,rCrD2rdt,rCrD20

rdt,rCrD2Sdt,rCrDrdt,rCrD 22

dD2rdt,rCdD2Sdt,rCrD2

AdDt2tr 2 for diffusion in d-dimensional space

= 1

Let’s consider diffusion of particles that are initially concentrated at the

origin of our coordination frame, -Dirac delta function r0,rC

Solution:

Dt

r

DttrC d 4

exp4

1,

2

2/


Side-note: Review of calculus used in previous page (not tested)

zz

yy

xx

Gradient:

Gradient of a scalar function: zz

yy

xx

)r(

Divergence of a vector function:z

F

y

F

x

F)r(F zyx

)r(F)r()r(F)r()r(F)r(

The divergence theorem: SV

Sd)r(Frd)r(F

2

2

2

2

2

22

zyx

Laplacian:

L’Hopital’s rule: if lim g(x)/f(x) result in the indeterminate form 0/0 orinf/inf, then

dxdf

dxdglim

)x(f

)x(glim


Thin film deposited on a semi-infinite piece of material

Using the same solution we have a different condition for

coefficient A:dx

4Dt

xexp

t

ASM

0

2

Using substitution of variable4Dt

xy

2

π4DASdyyexp

t

4DtASM

0

2

dy4Dtdx

D πS

MA

4Dt

xexp

Dt πS

Mtx,C

2

The solution is a half of a Gaussian concentration profile. At the surface, the boundary condition

t1

t2 > t1

is satisfied 0x

C

0x

x0

0x

C

0x


Analytical solutions: Pair of Semi-infinite Solids (I)

Think of the block of B as being made up of many thin layers ofthickness x and cross-sectional area S. Each layer initiallycontains C1Sx of B atoms. We know that if the areassurrounding a layer are initially B-free, the distribution of soluteatoms that originate from a thin layer can be described by theGaussian profile. Assuming that the presence of other solutes(atoms of type B) do not affect the diffusion of atoms thatoriginate from the layer, the overall solution can be found assuperposition of the Gaussian profiles from the layers.

A

C1

0t2 > t1

t1

B

0x x

x

C1

0

n

1i

2i1

4Dt

xxexp

Dt π2 S

SΔx Ctx,C

2x- 1x-4x- 3x-

M


Analytical solutions: Pair of Semi-infinite Solids (II)

x

C1

0

n

1i

2i1

4Dt

xxexp

Dt π2 S

SΔx Ctx,C

0

21 dw

4Dt

wxexp

Dt π2

Ctx,C

In the limit of n going to infinity and x going to zero we can go from the sum to an integral.

Substituting uDt2

wx

we can rewrite the integral:

0

u0

2

u1

2

u1 duedue π

Cdue

π

Ctx,C

222

Dtx

Dtx

Using definition of the error function:

z-erfz-erf and 1erfat account th into takingand

z

0

2 dyyexpπ

2zerf

Dt2xerf1

2

Ctx,C 1

2x- 1x-4x- 3x-


Analytical solutions: Constant surface composition (I)

A similar solution of the diffusion equation can be derived

for a constant surface concentration. If the initial

concentration of solute atoms in the specimen is C0 and

concentration at the surface is Cs, the analytical solution of

the Fick’s second law is

Dt2

xerfCC-CC 0ss

The error function is tabulated in books of standardmathematical functions and is defined as an internalfunction in some of programming languages.

erf (z) = 1 for z = erf (z) = 0 for z = 0erf (z) = 0.5 for z = 0.5

s0xCC

where erf is the error function defined by the following

equation: z

0

2 dyyexpπ

2zerf

0xCC

erf (z)1

-1

0.5

0.5


Summary

Make sure you understand language and concepts:

Numerical integration of the diffusion equation (not tested)

Finite difference method

Spatial and time discretization

Initial and boundary conditions

Stability criterion

Analytical solutions for special cases

plane source

thin film on a semi-infinite substrate

diffusion pair

constant surface composition

finite difference method spatial and time discretization...

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