Finite Difference Method (FDM) for Solving Ordinary and Partial Differential Equations - P, Lab Reports of Materials science

Download Finite Difference Method (FDM) for Solving Ordinary and Partial Differential Equations - P and more Lab Reports Materials science in PDF only on Docsity!

MAT E 460

ENGINEERING COMPUTATION LABORATORY

FINITE DIFFERENCES

Prof. Antonios Zavaliangos

LeBow 441, x2078, [email protected]

What is the

Finite Difference Method (FDM)

•^ Finite difference method is actually a set oftechniques for the numerical solution of ordinaryand partial differential equations. • It is “lighter” mathematically wrt finite elements • It is “easy” to implement computationally • It has problems with complex geometry...

Shooting Method

Once possibility is to assume that j(0)=j

and integrate bya

Euler Forward, Backward, R-K etc. In that case thecomputed value of c(L) may be c

which is possible that isa^

c^ a^

≠^ c

as the BC require. 0

In that case we need to adjust j

so that c(L) becomes ca^

This requires a number of iterations…This is the so called shooting method...

Finite Difference Method

A alternative is based on the FDM.Discretize the length L by using N(equi-spaced) points.Approximate the derivatives:Write the governing equation usingthe above approximations for each ofthe n-

inner

points

c^1

c^2

c^3

c^ n-

c^ n

2

1

1 2 2

1 1

1 1 2

x

c c c cd dx

c x c c x c c x c dc dx

i i i

i i i i i i^

−

−

−

1

2

4

3

2

3

2

1

1 1

2

1

1

^ ++ 

^ + 

^ ++ 

^ + 

^ ++ 

^ + 

−

−

−

−

n

n

n

i i i i i

c xv D

c xv D

c

c xv D

c xv D c

c xv D

c xv D c

c x cv D

x

c c c

PDE-FDM

x

t

0 1 2

…

i^

….^

n

0 1 2... j... m

δ x

δ t c t c

x

c c c D

ji ji ji ji ji

δ

δ

, (^1) ,

2

, 1 , , 1

−^

−

IC BC

tDδ< 2 x δ

Stability conditiono/w problems...

If I know i-1, i, i+1 at jI can determine i at j+

PDE-FDM

dc^ dt cd dy cd dx D^

^ 

+^

2 2 2 2

Extensions to 2-D:is tedious but computationally “easily” implemented3-D also...

Parabolic PDE

x

t

0 1 2

…

i^

….^

n

0 1 2... j... m

δ x

δ t

c t c

x

c c c D

ji ji ji ji ji

δ

δ

(^1) , (^1) ,

2

, 1 , , 1

−

−

+^

IC BC

dc^ dt cda dx 2 =^2 2

c t c

x

c c c a

ji ji ji ji ji

δ

δ

, (^1) ,

2

(^1) , 1 (^1) , (^1) , 1 2

−^

+−

Euler ForwardEuler Backward ++ Other methods exist...

Elliptic PDE

x

y

0 1 2

…

i^

….^

n

0 1 2... j... m

δ x

δ y

ji ji ji ji ji ji ji

f

y

c c c

x

c c c^

,

2

(^1) , , (^1) ,

2

, 1 , , 1

−^

−

− −

BC

), ( 2 2 2 2

yx f cd dy cd (^) dx

+^

e.g., steady state diffusion in 2-D

(n-2)

. (m-2) equations with.(n-2)^

(m-2) unknowns

BOTTOM LINE

•^

FDM provides an mathematically simple and easyto implement computationally method to solvehigher order ordinary and partially differentialequations.

•^

Solving complex geometry 2 or 3D problems canbe a challenge.