lecture 4 - taylor series expansion and finite difference method

8/6/2019 Lecture 4 - Taylor Series Expansion and Finite Difference Method

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Errors and Approximations

Analytical solutions are exact

Numerical solutions are approximate

Real-life models are so complex that analyticalsolutions may not be possible

Numerical methods provide an alternative route


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Error in the numerical solution

A formal definition:Given a mathematical model,

Error = |Exact Approximate|

= |Analytical Numerical|


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Determination of errors

Evidently, a good numerical methodshould have small errors!

To minimize error, we must understand

> the source of errors

> how the numerical method isconstructed


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Round-off error

In a computer, numbers are represented as (eg.)1.63 = 0.163E+01, or +0.8352167E+15(single precision)

Single precision: 10-38 to 10+37 with 7 decimalplaces

Double precision: 10-308 to 10+309 with 14 decimal

places

Irrational numbers such as 1/7 and will alwayshave round-off errors.

3


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Truncation error: an example

....

!

........

!32

132

++++++=

n

xxxxe

nx


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Truncation error

33.3

7.69

1.27

0.158

0.0158

39.3

9.02

1.44

0.175

0.0172

0.00142

1

1.5

1.625

1.645833333

1.648437500

1.648697917

1

2

3

4

5

6

a(%)t(%)Result (x=0.5)Terms

...!

...........!32

132

++++++=n

xxxxe

nx


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Relative error

Definition 1:

Definition 2:

Definition 3:

%100valuetrue

errortruet =

%100

ionapproximat

errorionapproximata =

%100.

..

approxcurrent

approxpreviousapproxcurrent

a

=


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Terminology

Round-off error

Truncation error

Discretization error

Relative error

Propagation of error

Condition number


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Taylor Series Expansion

........)(!

)(....

)(!3

)()(

!2

)(''))((')()(

)(

3)3(

2

++

++++=

n

n

axn

af

axaf

axaf

axafafxf

Taylor series approximation


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Taylor series with remainder term

dttfn

txR

asdefinedisRremainderthewhere

Raxn

af

axaf

axaf

axafafxf

n

x

a

n

n

n

n

n

n

)(!

)(

)(!

)(....

)(

!3

)()(

!2

)(''))((')()(

)1(

)(

3)3(

2

+

=

++

++++=

t is a dummy variable


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Properties of Taylor Series

Infinite series

Uniformly convergent if all derivatives exist

Note that the infinite series for exp(x), cos(x)

are special cases of Taylor series expansion.


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Zero order approximation

First order approximation

Second order approximation

Truncation of Taylor series

)()( 1 ii xfxf +

))((')()( 11 iiiii xxxfxfxf + ++

2

111 )(!2

)(''

))((')()( iii

iiiii xx

xf

xxxfxfxf ++ +++


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Example

Use zero-through fourth-order Taylor seriesexpansions to approximate the function

fromxi = 0 with h =1. That is, predict the

functions value atxi+1 = 1

2.125.05.015.1.0)(

234 +=xxxxxf


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An example


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A one dimensional grid

1ix ix

h

i-1 i i+1

1+ix


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Taylor series on a grid

Consider a set of pointsx(i) separated by adistance h.

1)1(

)(

3)3(

2'

1

)!1(

)(

)(

!

)(....

!3

)(

!2

)('')()()(

++

+

+=

=

++

++++=

nn

n

i

n

ni

n

iiiii

hn

fR

nowistermremainderthewhere

xix

Rhn

xf

hxf

hxf

hxfxfxf


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Useful estimates

Note that powers ofh are much smaller than

h.

Hence, for small values ofh, the infinite

series can be truncated.


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Order of approximation

If the nth order derivative is included in the

Taylor series, the approximation is said be

of order n.


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Applications of TSE

Extrapolation Numerical Differentiation

Error estimates

Error propagation

Condition number

Root finding


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Finite difference Formulas

forward difference

backward difference

central difference

where: h is the step size

O(h) and O(h2) are the truncation errors

)()()(

)(1

1' hOxx

xfxfxf

ii

iii +

=

+

+

)()()(

)( 1' hO

h

xfxfxf iii +

=

)(2

)()()( 211' hO

h

xfxfxf ii +

= +


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A structured grid


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Unstructured grid


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Complex geometry


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Solving a partial differential equation


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Construction of a system of algebraicequations


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Time dependent problems

Subjected to initial and

boundary conditions

One dimension


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Explicit and implicit approaches

explicit

Crank-Nicolson


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Implementation of an implicit technique

Matrix equation is solved

sequentially in time.


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Consistency

In the limit of the mesh size and time step

approaching zero, the algebraic equation

approaches the partial differential equation.


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Stability

The numerical algorithm does not amplifysmall errors in initial or boundary

conditions.

Von Neumann analysis: Look for conditionsunder which a periodic disturbance grows

exponentially fast.


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Stability of explicit and implicit schemes

Implicit schemes areunconditionally stable.

Explicit schemes are only

conditionally stable. Fourier number criterion:


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Pure advection equation


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Stability of an explicit scheme for theadvection equation

Courant number

criterion (also called theCFL condition)


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Advection-diffusion equation

Stability: Both Fourierand Courant number

conditions apply.


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Upwinding

Consider the solution of

the advection-diffusion

equation by the method fo

finite differences.


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Artificial diffusion

Upwinding increases effective viscosity(conductivity in transport problems).

Oscillations in the numerical solution are

damped.

Need to find a compromise between a

damped solution and an accurate solution.


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Closure

Finite difference method stems from the Taylorseries expansion.

It converts a differential equation to a difference(algebraic) equation.

Several numerical issues need to be addressed:errors,

consistency,

stability,oscillations,

computational effort.


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THANK YOU!

lecture 4 - taylor series expansion and finite difference method

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