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Finite Difference Approximations-Numerical Methods in Engineering-Lecture 14 Slides-Civil Engineering and Geological Sciences, Slides of Numerical Methods in Engineering

University of Notre Dame du Lac (ND)Numerical Methods in Engineering

Practical issues in applying finite difference approximations. Finite Difference Approximations, Variable Coefficients, Two Dimensional, Finite Difference, Spatial Index, Examples

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2011/2012

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CE 341/441 - Lecture 14 - Fall 2004

p. 14.1

LECTURE 14

PRACTICAL ISSUES IN APPLYING FINITE DIFFERENCE APPROXIMATIONS

Derivatives of Variable Coefficients

• Consider the term

where is a coefficient which is a function of

• Proceed in steps. First let:

• Now evaluate:

------gx()

df x()

-------------





ugx()

------

≡

------ui1()

Partial preview of the text

Download Finite Difference Approximations-Numerical Methods in Engineering-Lecture 14 Slides-Civil Engineering and Geological Sciences and more Slides Numerical Methods in Engineering in PDF only on Docsity!

CE 341/441 - Lecture 14 - Fall 2004

p. 14.

LECTURE 14PRACTICAL ISSUES IN APPLYING FINITE DIFFERENCE APPROXIMATIONSDerivatives of Variable Coefficients • Consider the term

where

is a coefficient which is a function of

• Proceed in steps. First let:• Now evaluate:

d ------^ dx

g x (^

df ) x (^

-------------^ dx ^

^

g x (^

df ) ----- dx

≡ du ------^ dx

u^ i

(^1) ( )

CE 341/441 - Lecture 14 - Fall 2004

p. 14.

• Now evaluate

and

hi i-

i-1/

i+1/

i+ hi+

(^1) ( (^) u i )

u^ i^

1 ---+ 2

u^ i^

1 ---– 2

1 ---^2 h^ i^

h^ i^

(^1) +

u^ i^

1 ---+ 2

u^ i^

1 ---– (^2) u i^

1 ---+ 2

g^ i^

f^ 1 ---+ 2 i^ 1 1 ---+^2 (^ )^

g^ i^

1 ---+ 2

f^ i^

(^1) +

f^ – i h^ i^

(^1) + ---------------------- 

^

^

=^

u^ i^

1 ---– 2

g^ i^

f^ 1 ---– 2 i^ 1 1 ---–^2 (^ )

g^ i^

1 ---– 2

f^ i^

(^1) –

-^ h^ i ---------------------

^

^

^

=^

CE 341/441 - Lecture 14 - Fall 2004

p. 14.

Two Dimensional Finite Difference Approximations • Define nodes in a two-dimensional plane with

•^

= spatial index in the

-direction

•^

= spatial index in the

-direction

fi^ , j+2i, j+

i+2, j+

f i, j+1 i, j-1 i, j-

fi^ +2, j

∆ y^

∆ x

i-2, j

i-1, j

i, j i+1, j

i+2, j

fi^ , j

fi+

2 , j+

CE 341/441 - Lecture 14 - Fall 2004

p. 14.

• When taking partial derivatives w.r.t.

, we hold

constant

and

• Similarly

∂ f ------ ∂ x

i^ j ,

f^ i^

1 j ,+

f^ i^

1 j ,–

2 ∆ x

f ∂ x

2

-^ i^

j ,

f^ i^

1 j ,+

2 f^

i^ j ,^

f^ i^

1 j ,–

∆ x

f ∂ y

2

-^ i^

j ,

f^ i^

(^1) + ,^

2 f^

i^ j ,^

f^ i^

j^^1 – ,

∆ y

CE 341/441 - Lecture 14 - Fall 2004

p. 14.

Example 2 • Let’s consider: • Draw a molecule diagram for the 2nd order central difference formulation.• Obtain

by finding

⇒ ⇒

2 f (^

)^

f ∂ x

4 -^

f ∂ x

(^2) y

-^

f ∂ y

4 +^

=^

f ∂ x

4 -^

(^2) ∂--------^2 ∂ x

2 f ---------^2 ∂ x ^

^

^

f ∂ x

4 -^

(^2) ∂--------^2 ∂ x

2 f ---------^2 ∂ x ^

^

^

f ∂ x

4 -^

(^2) ∂--------^2 ∂ x

(^1) ---------^2 ∆ x f^

i^ 1 j ,+

2 f i^ j ,^

f^ i^

(^1) – j ,

^

^

CE 341/441 - Lecture 14 - Fall 2004

⇒ ⇒ p. 14.

f ∂ x

4 -^

(^1) ---------^2 ∆ x

(^2) --------^2 ∂ x

f^ i^

1 j ,+ (^

)^

(^2) --------^2 ∂ x

f^ i^

j , (^

)^

(^2) ∂--------^2 ∂ x f^

i^^1 –^

j , (^

^

^

^

f ∂ x

4 -^

(^1) ---------^2 ∆ x

(^1) ---------^2 ∆ x f^

i^ 2 j ,+

2 f i^

(^1) + j ,^

f^ i^

j ,

(^

–^

(^2) ---------^2 ∆ x f^

i^ 1 j ,+

2 f^

i^ j ,^

f^ i^

(^1) – j ,

(^1) ---------^2 ∆ x

i^ j ,^

2 f i^

(^1) – j ,^

f^ i^

(^2) – j ,

(^

+^

f ∂ x

4 -^

(^1) ---------^4 ∆ x

i^ 2 j ,–

4 f i^

1 j ,–

6 f i^ j ,^

4 f i^

1 j ,+

f^ i^

2 j ,+

(^

(^14) ∆ x^

CE 341/441 - Lecture 14 - Fall 2004

p. 14.

• Evaluation of the mixed derivative:

⇒ ⇒ (^4) ∂ f ∂ x

(^2) y

-^

(^2) ∂--------^2 ∂ y

2 f ---------^2 ∂ x ^

^

^

f ∂ x

(^2) y

-^

(^2) ∂--------^2 ∂ y

(^1) ---------^2 ∆ x f^

i^^1

j ,–

2 f^

i^ j ,^

f^ i^

1 j ,+

^

^

f ∂ x

(^2) y

-^

(^1) ---------^2 ∆ x

(^1) ---------^2 ∆ y f^

i^ 1 j

^1 – ,

2 f^

i^ 1 j ,–

f^ i^

1 j

^1 +,

(^

–^

(^2) ---------^2 ∆ y f^

i^ j^

(^1) – ,^

2 f^

i^ j ,^

f^ i^

j^ (^1) + ,

(^1) ---------^2 ∆ y f^

i^ 1 j

^1 – , +^

2 f^

i^ 1 j ,+

f^ i^

1 j

^1 +,

(^

CE 341/441 - Lecture 14 - Fall 2004

⇒ p. 14.

• This then produces the following module

f ∂ x

(^2) y

-^

∆ x

(^2) y

-^ f

1 j

^1 – ,

2 f^

i^ 1 j ,–

f^ i^

1 j ^1 +,

[

2 f i^ j

^1 – ,

4 f i^ j ,^

2 f i^ j

^1 +,

f^ i^

1 j

^1 – , +^

2 f^

i^ 1 j ,+

f^ i^

1 j ,+

]

1^ ∆ y

(^2) ∆ x

CE 341/441 - Lecture 14 - Fall 2004

p. 14.

• To obtain

, we must add up modules:

• If

then:

f ∂ x

4 -^

f ∂ x

(^2) y

-^

f ∂ y

4 +^

∆ x

∆ y

-8 20 -8^1

-^ -

(^14) h

Finite Difference Approximations-Numerical Methods in Engineering-Lecture 14 Slides-Civil Engineering and Geological Sciences, Slides of Numerical Methods in Engineering

Related documents

Partial preview of the text

Download Finite Difference Approximations-Numerical Methods in Engineering-Lecture 14 Slides-Civil Engineering and Geological Sciences and more Slides Numerical Methods in Engineering in PDF only on Docsity!

LECTURE 14PRACTICAL ISSUES IN APPLYING FINITE DIFFERENCE APPROXIMATIONSDerivatives of Variable Coefficients • Consider the term

where

is a coefficient which is a function of

• Proceed in steps. First let:• Now evaluate:

^

(^1) ( )

• Now evaluate

and

^

^

=^

^

^

^

=^

Two Dimensional Finite Difference Approximations • Define nodes in a two-dimensional plane with

•^

= spatial index in the

-direction

•^

= spatial index in the

-direction

• When taking partial derivatives w.r.t.

, we hold

constant

and

• Similarly

2 ∆ x

2

2

Example 2 • Let’s consider: • Draw a molecule diagram for the 2nd order central difference formulation.• Obtain

by finding

)^

4

-^

(^2) y

-^

4

+^

=^

4

-^

^

^

4

-^

^

^

4

-^

^

^

4

-^

)^

)^

^

^

^

4

-^

(^

–^

(^

+^

4

-^

(^

• Evaluation of the mixed derivative:

(^2) y

-^

^

^

(^2) y

-^

^