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Review of Numerical Methods-Numerical Methods in Engineering-review3-Civil Engineering and Geological Sciences, Study notes of Numerical Methods in Engineering

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

Describe a function by passing a polynomial through a set of functional values and/or functional derivative values. Review of Numerical Methods, ODE Classification, Runge Kutta Formulae, Deriving Runge Kutta Methods, Multi Step Methods, Open Formulae, Closed Formulae, Predictor Corrector Methods

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CE 341/441 - Review 3 - Fall 2004

p. R3.1

REVIEW NO. 3

O.D.E. CLASSIFICATION

•I.V.P.’s

•B.V.P.’s

• Can always decompose an nth order i.v.p. into simultaneous 1st order i.v.p.’s

• e.g. - Converting the stated second order i.v.p. into two coupled first order i.v.p.’s

Ad2y

dt2

-------- Bdy

------Cy++ gt()=y0() yo

=dy

------0() Vo

Ad2y

dx2

-------- Bdy

------Cy++ gx()=y0() yo

=yL() yL

------z=y0() yo

Adz

----- Bz–Cy–gt()+= z0() Vo

Partial preview of the text

Download Review of Numerical Methods-Numerical Methods in Engineering-review3-Civil Engineering and Geological Sciences and more Study notes Numerical Methods in Engineering in PDF only on Docsity!

p. R3.

REVIEW NO. 3O.D.E. CLASSIFICATION • I.V.P.’s • B.V.P.’s • Can always decompose an

n

th^

order i.v.p. into

simultaneous 1

st^

order i.v.p.’s

• e.g. - Converting the stated second order i.v.p. into two coupled first order i.v.p.’s

A

2 y^2 dt

B

dy ----- dt

-^

g t

)^

y^ o

dy ----- dt

)^

V

A

2 y d x

2

B

dy ----- dx

-^

g x

(^

)^

y^ o

y L (

)^

y^ L

dy ----- dt

-^

z

)^

y^ o

A

dz ---- dt

-^

g t

)^

V

p. R3.

I.V.P. SOLUTIONS - 1ST ORDER EQUATIONS - SINGLE STEP METHODS • Solve

• Runge-Kutta formulae are single step methods^ INSERT FIGURE NO. 113 Runge-Kutta Formulae

(slope at

dy ----- dt

-^

y t ,(

)^

y^ o

yj+ tj+

tj yj

∆

1 +^

t a

g 1

g 2 2

a^ n

g n

(^

g^1

t^ j

j , (^

t^ j

j , (^

g^2

t^ j

t y

j ,^

p^2

(^

g^3

t^ j

t y

j ,^

p^4

(^

p. R3.

Step 2 • Substitute in Taylor series expanded forms of

into the R.K. formula

Step 3 • Taylor series expand for

about

is a function of

only

→

apply a 1-D Taylor series

• However by definition• Must use chain rule to differentiate

⇒

g^2

,^

1 +^

y t ( )

1 +^

dy ----- dt

(^2) t 2

-^

2 y^2 dt

O

t (^

dy ----- dt

-^

t y ,(

t y t

(^

2 y^2 dt

∂ f^ ∂

t

f ∂

y

dy ----- dt

2 y^2 dt

f ∂ t

f ∂

y

p. R3.

• Substituting Step 4 • Match term by term expressions for

obtained in steps 2 and 3

• Set up equations for coefficients and solve to a free variable Step 5 • Select the free variable

1 +^

t f

(^2) t 2

-^

f ∂ t

f ∂

y

^

^

^

O

(^3) t (^

p. R3.

SOLUTIONS TO I.V.P.’S - MULTI-STEP METHODS • Multi-step methods may involve more than one previous discrete point^ INSERT FIGURE NO. 114 • Open formulae, closed formulae and predictor-corrector methods are all multi-step

methods

yj+1 tj+

yj-

tj-

p. R3.

Open Formulae • Solve

such that unknown solution is computed explicitly (i.e.

is expressed in terms of only known values of

,^

, etc.)

How to Derive Open Formulae Step 1 • Take a forward Taylor series of

about

• However

⇒

etc.

• Substituting

dy ----- dt

-^

t y ,(

)^

y^ o

1 +^

t^ j

1 +^

dy ----- tdt

(^2) t 2!

-^

2 y

j dt

2 (^3) t 3!

-^

3 y

j dt

3

-^

dy ----- dt

2 y^2 dt

˙ f j

1 +^

t -----^ 2!

˙ j

t (^

-^

˙˙ fj

^

^

p. R3.

Advantages of Open formulae •^

only needs to be evaluated at

known

values, therefore no iteration is required

• More efficient per time step than R.K. Disadvantages of Open formulae • Stability• Accuracy• Difficult to change time step

CE 341/441 - Review 3 - Fall 2004

p. R3.

Closed Formulae • Solve

such that the unknown solution is computed implicitly

(i.e.

is expressed in terms of both unknown

and known values of

,^

, etc.)

How to Derive Closed Formulae Step 1 • Use a backward Taylor series expansion for

about

• Re-arrange and substitute

,^

etc.

dy ----- dt

-^

t y ,(

)^

y^ o

1 +^

y t ( )

y t

1 +^

dyt ----- dt

–^

(^2) t 2!

d -

2 y^2 dt

(^3) t 3!

d -

3 y^3 dt

H

. O

. T

dy ----- dt

2 y^2 dt

˙ f j^

1 +^

t 2

˙ f j^

(^2) t 3!

-^

˙˙ f j^

1 +^

p. R3.

Predictor-Corrector Methods Predictor

→

Open Formula

• e.g.

Corrector

→

Closed Formula

• e.g.

• first iteration

→

use predictor value

• then iterate until convergence

Starter • Applies a R.K. formula to start the computation

1 (^0) ( ) +

3 ---^2

1 ---^2

1 k^ +

t 2

1 k ( )^ +

[^

]

p. R3.

Modifier • Improves first iteration value for the Corrector based on estimated truncation error Advantages of Predictor-Corrector Methods • Very efficient

• Less work per time step as compared to R.K. methods• Only few iterations needed when compared to closed formulae

• Good stability (stability of the corrector)• Very accurate (more so than open formulae)• Easy to estimate errors

p. R3.

COURSE SUMMARYSurvey of Many Numerical Methods Numerical Interpolation • Describe a function by passing a polynomial through a set of functional values and/or

functional derivative values

• Example Uses:

• Find functional values at locations other than the nodes• Basis of the finite element method

Numerical Differentiation • Find discrete approximations to differentiation. Derivatives are approximated by sums

and differences of functional values at discrete points (nodes) in space/time

• Uses:

• Solve o.d.e.’s and p.d.e.’s using F.D. methods• Estimate errors in numerical approximations

p. R3.

Numerical Integration • Find integrals based on discrete functional values at a given set of integration points• Uses:

• Integrate results such as flows etc.• Use for finite element methods (integral methods)

O.D.E./I.V.P.’s

→

Time Dependent Problems

• Can reduce any order o.d.e./i.v.p. into a system of 1st order o.d.e./i.v.p.’s• Time march

→

go from one time level to the next

• One step methods

→

Runge-Kutta

• Multi-step methods

→

Open, Closed and Predictor-Corrector Methods

O.D.E./B.V.P.’s

→

Spatially Dependent Problems

• Solve by substituting in F.D. approximations for the terms• Generate enough algebraic equations to solve for the unknowns

p. R3.

Course Concepts • Many of these methods are inter-related/used together

• In their derivation• In their use

• All methods are based on representing continuous functions at discrete points in

space/time

• As engineers we wish to solve complicated problems often described by mathematical

equations

• Solve o.d.e.’s/p.d.e.’s• Numerical methods allow us to solve mathematical problems for which closed form

solutions are not available

p. R3.

Review of Numerical Methods-Numerical Methods in Engineering-review3-Civil Engineering and Geological Sciences, Study notes of Numerical Methods in Engineering

Related documents

Partial preview of the text

Download Review of Numerical Methods-Numerical Methods in Engineering-review3-Civil Engineering and Geological Sciences and more Study notes Numerical Methods in Engineering in PDF only on Docsity!

REVIEW NO. 3O.D.E. CLASSIFICATION • I.V.P.’s • B.V.P.’s • Can always decompose an

n

order i.v.p. into

simultaneous 1

order i.v.p.’s

• e.g. - Converting the stated second order i.v.p. into two coupled first order i.v.p.’s

A

2 y^2 dt

B

-^

)^

y^ o

)^

V

A

2

B

-^

(^

)^

y^ o

)^

y^ L

-^

z

)^

y^ o

A

-^

)^

V

I.V.P. SOLUTIONS - 1ST ORDER EQUATIONS - SINGLE STEP METHODS • Solve

• Runge-Kutta formulae are single step methods^ INSERT FIGURE NO. 113 Runge-Kutta Formulae

(slope at

-^

)^

y^ o

(^

(^

(^

Step 2 • Substitute in Taylor series expanded forms of

into the R.K. formula

Step 3 • Taylor series expand for

about

is a function of

only

apply a 1-D Taylor series

• However by definition• Must use chain rule to differentiate

,^

(^2) t 2

-^

2 y^2 dt

O

-^

(^

2 y^2 dt

t

y

2 y^2 dt

f ∂ t

y

• Substituting Step 4 • Match term by term expressions for

obtained in steps 2 and 3

• Set up equations for coefficients and solve to a free variable Step 5 • Select the free variable

(^2) t 2

-^

f ∂ t

y

^

^

^

O

SOLUTIONS TO I.V.P.’S - MULTI-STEP METHODS • Multi-step methods may involve more than one previous discrete point^ INSERT FIGURE NO. 114 • Open formulae, closed formulae and predictor-corrector methods are all multi-step

methods

Open Formulae • Solve

such that unknown solution is computed explicitly (i.e.