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Approximation for Solving Ordinary Differential Equations, Slides of Numerical Methods in Engineering

Iterative solution of algebraic equations puts the nonlinear term on the r.h.s. and iterates until convergence. There may be convergence problems. Approximation of Ordinary Differential Equations, Initial Value Problems, Boundary Value Problems, Initial Value Problems, Euler's Method, Time Stepping, Time Marching

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CE 341/441 - Lecture 15 - Fall 2004
p. 15.1
LECTURE 15
APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY
DIFFERENTIAL EQUATIONS
Ordinary Differential Equations
Initial Value Problems
For Initial Value problems (IVP’s), conditions are specified at only one value of the
independent variable initial conditions (i.c.’s)
For example a simple harmonic oscillator is described by
= location dependent variable
= time independent variable
Ad2y
dt2
-------- Bdy
dt
------Cy++ gt()=y0() yo
=dy
dt
------0() Vo
=
y
t
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

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CE 341/441 - Lecture 15 - Fall 2004

p. 15.

LECTURE 15APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARYDIFFERENTIAL EQUATIONSOrdinary Differential Equations Initial Value Problems • For Initial Value problems (IVP’s), conditions are specified at only one value of the

independent variable

initial conditions (i.c.’s)

  • For example a simple harmonic oscillator is described by -^

= location

dependent variable

-^

= time

A independent variable

d

2 y^2 dt

B

dy ----- dt

-^

Cy

g t

y^

)^

y^ o

dy ----- dt

)^

V

o

=

y t

CE 341/441 - Lecture 15 - Fall 2004

p. 15.

Boundary Value Problems • For Boundary Value Problems (BVP’s) conditions are specified at two values of the

independent variable (which represent the actual physical boundaries)

  • Example General Initial Value Problems • Any IVP can be represented as a set of one or more 1st order d.e.’s each with an i.c.• Example
    • Let

and we can develop a system of 2 first order O.D.E.’s which are coupled

d

2 y d x

2

D

dy ----- dx

-^

Ey

h x

(^

y^

)^

y^ o

y L (

)^

y^ l

A

d

2 y^2 dt

B

dy ----- dt

-^

C

g t

y^

)^

y^ o

dy ----- dt

)^

v^ o

z^

dy ----- dt

dy ----- dt

-^

z =

y^

)^

y^ o =

dz ----- dt

B ---- A

–^

z^

C ---- A

–^

g t

( ) A

z^

)^

v^ o =

CE 341/441 - Lecture 15 - Fall 2004

p. 15.

Solution to a 1st Order Single Equation IVP

with specified i.c.

Euler Method • The Euler method is a 1st order method• We evaluate the o.d.e. at node

and use a forward difference approximation for

dy ----- dt

-^

f^

y t ,(

y t

o (^

)^

y^ o

j^

dy ----- dt

j

dy ----- dt

j

f^

y^

, j t^ j (^

y^

j^

1 +^

y^

j

t


-^

f^

y^

j^

t^ , j

(^

y^

j^

1 +^

y^

j^

t f

y^

j^

t^ , j

(^

CE 341/441 - Lecture 15 - Fall 2004

p. 15.

- Simply “march” forward in time - From time level

(time =

  • To time level

(time =

  • We note that

equals the slope at

  • Therefore to obtain

simply add

to

j^

t^ j

j^

t^ j

1 +^

t^ j

t

yj+

yj

tj^

tj+

t f (t

, yj

)j

f^

t^ j

y^

j , (^

)^

t^ j

y^

j^

1 +^

t f

t^ j

y j , (^

)^

y^

j

CE 341/441 - Lecture 15 - Fall 2004

p. 15.

  • Discretize the o.d.e. at a general node• Approximate

using a forward difference approximation

Next Value = Previous Value + Run

Slope

  • Equation relates a known time level

to the new time level

.^

This process is known

as “time stepping” or “time marching”

i dy ----- dt

i

y^ i

y i

=

dy ----- dt

i

y^ i

1 +^

y^ i

t


-^

y^ i

y i

=

y^ i

1 +^

y^ i

t y

i^

y^ i

×

i^

i^

CE 341/441 - Lecture 15 - Fall 2004

p. 15.

  • The i.c. indicates that• Take 1st time step

  • Note that

where

in this case

at

  • Take next time step

at

y^ o

= i

i^

t^ i

i^

t^

t^ o

⋅^

i

t

t^ o

y^1

y^ o

ty

o^

y^ o

y^1

×

×

y^1

t^1

i^

i^

y^2

y^1

t y

1

y^1

y^2

×

×

y^2

t^2

CE 341/441 - Lecture 15 - Fall 2004

p. 15.

  • Take next time step

at

  • Take next time step

at

  • We can continue time marching

t

Numerical Solution using

and the Euler

Method

Numerical Solution using

and the Euler

Method

Exact

Solution

1.0000 (i.c.)

1.0000 (i.c.)

y^5

t^5

y^6

t^6

t^

t^

CE 341/441 - Lecture 15 - Fall 2004

p. 15.

General Observations for Solving IVP’s • Solution to o.d.e.’s can be very simple using finite difference approximations to repre-

sent differentiation

  • Accuracy is dependent on the time step

! We need to understand the error behavior

  • As

, the solution gets better

  • IVP’s are solved using a time marching process

Begin at one end and march forward

up to the desired point or indefinitely

  • At each time step, we introduce a new unknown,

, which is solved for by writing

and solving the discrete form of the IVP at node

j.

Solutions to Boundary Value Problems • Boundary value problems must be 2nd order o.d.e.’s or higher• We apply FD approximations to the various terms in the differential equation to obtain

discrete approximations to the differential equations at points in space.

  • Unknown functional values at the nodes will be coupled and require the solution of a

system of simultaneous equations

matrix methods.

t

t^

y^

j^

1

CE 341/441 - Lecture 15 - Fall 2004

p. 15.

  • Therefore we generate

equations (1 for each interior point and 1 for each boundary

node) to solve for the

unknowns:

... ...

n^

n^

y^0

y^2

y^1

-^

y^0

A

x (^

y

1

B

x (^

y^3

y^2

-^

y^1

A

x (^

y

2

B

x (^

y^

j^

1 +^

y^

j

-^

y^

j^

1

-^

A

x (^

y^

j

B

x (^

y^ n

y^ n

1

-^

y^ n

2

–^

A

x (^

y^

n^

1

B

x (^

y^ n

1 +^

y^ n

-^

y^ n

1

-^

A

x (^

y^

n

B

x (^

y^ n

1 +^

CE 341/441 - Lecture 15 - Fall 2004

p. 15.

  • Collect coefficients of unknowns and write in matrix form:where

α

α

α

.^

.^

.^

.^

.^

.^

α

α

α

y^0 y^1 y^2 y^3 y^ n

1

  • y n y^ n

1

B

x (^

B

x (^

B

x (^

B

x (^

B

x (^

α

–^

A

x (^

CE 341/441 - Lecture 15 - Fall 2004

p. 15.

  • Solution strategies include:
    • Iterative solution of algebraic equations puts the nonlinear term on the r.h.s. and iter-

ates until convergence. There may be convergence problems.

  • Linearization of the nonlinear terms. Use Taylor series to approximate the nonlinear

terms.