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CS537 Numerical Analysis Lecture Numerical Solution of Ordinary Differential Equations Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 402060046 April 15, 2010
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Page 1: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

CS537

Numerical Analysis

Lecture 

Numerical Solution of Ordinary Differential EquationsNumerical Solution of Ordinary Differential Equations

Professor Jun ZhangProfessor Jun Zhang

Department of Computer ScienceUniversity of Kentuckyy y

Lexington, KY 40206‐0046

April 15, 2010

Page 2: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

What is ODE

An Ordinary Differential Equation (ODE) is an equation that involves one or more derivatives of an unknown function

A solution of a differential equation is a specific function that satisfies the equation

For the ODE

The solution is

texdtdx

=−The solution is

where c is a constant.  Note that the solution is not unique

tt cetetx +=)(q

Also, the function has derivatives only respect to one variable

2

For derivatives with multiple variables, the equation is usually called partial differential equation (PDE)

Page 3: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

An Example of ODE p

A ODE l ti f ti f th ti t

3

An ODE solution as a function of the time t

Page 4: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Initial‐Value Problem

A differential equation does not, in general, determine a unique solution f i Addi i l di i d d d i ifunction. Additional conditions are needed to determine a unique solution

The initial value problem has the following standard formThe initial‐value problem has the following standard form

⎩⎨⎧

==

saxxtfx

)(),('

Here x is a function of t, x(a) is the given initial value of the problem. We would like to determine the value of x at any time before or after a

Some examples of initial value problems and solutions

SolutionValue InitialEquationt

4436)1(16'10)0(1'

2 +−==−=−==+=ttxxtx

exxxx t

Page 5: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Numerical Solution

Analytical solutions in closed forms are only available for special differential equationsdifferential equations

Most differential equations encountered in practical applications do not have analytical solutions

Some analytical solutions may be too complicated to be useful in practice

A numerical solution of a differential equation is usually obtained in the form of a table of discrete values

Interpolation procedures can be used to obtain all values of theInterpolation procedures can be used to obtain all values of the approximate numerical solution within a given interval

A large amount of data may be displayed as a solution curve on a color

5

A large amount of data may be  displayed as a solution curve on a color graphical monitor

Page 6: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

ODE and Intergration

Consider the differential equation

⎪⎩

⎪⎨⎧ =

sax

xrfdrdx

)(

),(

We integrate it from t to t+h

⎪⎩ = sax )(

∫∫++

=htht

drrxrfdx ))((

We obtain

∫∫ =tt

drrxrfdx ))(,(

∫+

+=+ht

drrxrftxhtx ))(()()(Replacing the integral with one of the numerical integration rules we studied before, we obtain a formula for solving the differential equation

∫+=+t

drrxrftxhtx ))(,()()(

6

, g q

Page 7: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Numerical Rules

Using the left rectangular approximation formula, we have

∫+ht

which leads to the Euler’s method

∫+

≈ht

ttxthfdrrxrf ))(,())(,(

On the other hand, the trapezoid rule leads to 

))(,()()( txthftxhtx +=+

and the implicit formula

))](,())(,([2

))(,( htxhtftxtfhdrrxrfht

t+++≈∫

+

and the implicit formula, 

))](,())(,([2

)()( htxhtftxtfhtxhtx ++++=+

7

because x(t+h) appears on both sides

Page 8: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Taylor Series Methodsy

The Taylor series of a function 

+++=+ )('"1)(''1)(')()( 32 txhtxhthxtxhtx

h l l f ( h) h h f

LL ++++

+++=+

)(!

1)(!4

1

)(!3

)(!2

)()()(

)()4(4 txhm

txh

txhtxhthxtxhtx

mm

gives the numerical solution of x(t+h) when we truncate the series after the (m+1)st terms

If h is small and if we knowIf h is small, and if we know

the Taylor series computes very accurate value of x(t+h)

)(,),("),('),( )( txtxtxtx mKthe Taylor series computes very accurate value of x(t+h). 

If we truncate the series after the (m+1)st term, the method is called the Taylor series method of order  m

8

y

The Taylor series method of order 1 is known as the Euler’s method

Page 9: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Euler’s Method

The initial value problem

⎧ ttf ))(('

over the interval [a,b], we use the first two terms of the Taylor series⎩⎨⎧

==

axaxtxtfx

)())(,('

over the interval [a,b], we use the first two terms of the Taylor series

This gives the formula of the Euler’s method

)(')()( thxtxhtx +≈+

The computation starts from t=a and stops at t=b with n steps of size

))(,()()( txthftxhtx +=+

nabh )( −

=

9

Page 10: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Euler’s method

Using Euler’s method to solve an ODE. The typical behavior of the computed solution is that it runs away from the exact solution as more steps are taken, due to the accumulation of errors at each step

10

Page 11: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Taylor Series Method of Higher Ordery g

Consider the initial‐value problem

1' 32 ++= txx

d ff h l

4)1(1

−=++=

xtxx

We differentiate the equation several times, as

3'2"1'

2

32 ++=

ttxx

6''2"2'"3'2"

)4(

2

++=+=

txxxxxtxxx

These terms can be applied in order in the Taylor series method and the computed solution will be more accurate. For example x(2)=4.235841 

6"'6'"2)4( ++= xxxxx

11

p p ( )from the Euler’s method. x(2)=4.3712096 from the use of the 4th

derivative. The exact value to five significant figures is x(2)=4.3712 

Page 12: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Type of Errorsyp

When we truncate the Taylor series, we introduce the local truncation error  1 55

The truncation error is of order h5 , or O(h5).

)(120

1 55 ξxh

The second type of error is the accumulation effects of all local truncation errors

The computed value of x(t+h) is in error because x(t) is already in error, due to the previous truncation error, and the current step involves another truncation erroranother truncation error

The roundoff error can also contribute to the accumulation of error

12

These errors may be magnified by succeeding steps

Page 13: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Runge‐Kutta MethodsThe high order Taylor method for solving the initial‐value problem

⎩⎨⎧

==

xaxxtfx

)(),('

needs the derivatives of x by differentiating the function f . This is not convenient. We need some highly accurate methods that make use of fdirectly

⎩ = axax )(

directly 

The Runge‐Kutta methods are a class of methods using multiple evaluations of f, not its derivatives, to enhance computational accuracyevaluations of f, not its derivatives, to enhance computational accuracy

We will illustrate the procedure to derive the Runge‐Kutta method of order 2, and give the formula for Runge‐Kutta method of order 4, which g gis popularly used

We assume that f(t,x) can be computed for any (t,x), so that x(t), x(t+h), 

13

x(t+2h), …, can be computed

Page 14: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Taylor Series for f(x,y)y f( ,y)

We can expand the Taylor series in two variables1

i∞ ⎞⎛ ∂∂

The first few right hand terms are

),(!1),(

0yxf

yk

xh

ikyhxf

i∑∞

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=++

The first few right‐hand terms are

),(0

fyxfy

kx

h =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

),(1

yfk

xfhyxf

yk

xh

yx

∂∂

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

⎟⎠

⎜⎝ ∂∂

2),( 2

22

2

2

22

2

yfk

yxfhk

xfhyxf

yk

xh

yxyx

∂∂

+∂∂

∂+

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂⎠⎝ ∂∂

14

M yyy ⎠⎝

Page 15: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Taylor Series for f(x,y)y f( ,y)

The Taylor series can be truncated as

We now use subscripts to denote the partial derivatives

),(!

1),(!

1),(1

0yxf

yk

xh

nyxf

yk

xh

ikyhxf

nin

i⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=++ ∑−

=

We now use subscripts to denote the partial derivatives

( )+++

++=++

21

)(),(

22yyxyxx

yx

fkhkffh

kfhffkyhxf

( )

( )++++

+++

33!3

1

2!2

3223yyyxyyxxyxxx

yyxyxx

fkfhkkfhfh

fkhkffh

L+ L++++=+ xxxxxx fhfhhffyhxf

!3!2),(

32

15L++++=+ yyyyyy fkfkkffkyxf

!3!2),(

32

Page 16: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Runge‐Kutta Method of Order 2Runge Kutta Method of Order

Define two function evaluations

⎧ )(hfK

We add the linear combination of these quantities to the value of x at t to⎩⎨⎧

++==

),(),(

12

1

KxhthfKxthfK

βαWe add the linear combination of these quantities to the value of x at t to obtain the value at t+h

or 2211)()( KwKwtxhtx ++=+

We want to determine the constants in the above expression so that it 

)),(,(),()()( 21 xthfxhthfwxthfwtxhtx βα ++++=+

will be as accurate as possible

In other words, we want the above expression to match the Taylor series t t ibl

16

to as many terms as possible

Page 17: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Euler’s Method & Runge‐Kutta Methodg

The Euler’s method (left) requires one function evaluation at each step

17

The second‐order Runge‐Kutta method (right) requires two function evaluations at each step

Page 18: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Runge‐Kutta Method of Order 2g

Compare the intended formula

))(()()()( hfhhfhfh βwith the Taylor series

)),(,(),()()( 21 xthfxhthfwxthfwtxhtx βα ++++=+

)('"1)("1)(')()( 2hhh

If we set w1=1 and w2=0, the two expressions agree to the h term

L++++=+ )('"!3

)("!2

)(')()( 2 txtxhthxtxhtx

1 2

The formula obtained is

)()()( xthftxhtx +=+which is the Euler’s method, since

),()()( xthftxhtx +=+

)()(' xtftx =

18

),()( xtftx =

Page 19: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Runge‐Kutta Method of Order 2g

Revisit the two variable  Taylor series

1 2⎞⎛ ∂∂

Using this formula in

),(21),(

2

yxfx

hft

ahhffhffhfxhtf xt ⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+++=++ ββαβα

Using this formula in

We have

)),(,(),()()( 21 xthfxhthfwxthfwtxhtx βα ++++=+

Note that

)()()()( 322

2221 hOffhwfhwhfwwtxhtx xt +++++=+ βα

ffftx

xf

dtdt

tf

dtxtdf

dtdxx xt +=⎟

⎠⎞

⎜⎝⎛∂∂

⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛∂∂

===),('"

19

txdttdtdt ⎠⎝ ∂⎠⎝ ∂⎠⎝⎠⎝ ∂

Page 20: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Runge‐Kutta Method of Order 2g

The Taylor series equation11

becomes 11

L++++=+ )('"!3

1)("!2

1)(')()( 32 txhtxhthxtxhtx

becomes

Compare the two expressions of x(t+h), we have

)(21

21)( 322 hOffhfhhfxhtx xt ++++=+

One solution is21

21 1 2221 ===+ wwww βα

Oth l ti ibl b t th t ti ill b th

21

21 1 1 21 ==== wwβα

20

Other solutions are possible, but the truncation errors will be the same

Page 21: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Runge‐Kutta Method of Order 2g

So the Runge‐Kutta Method of Order 2 ishh

Or equivalently 1

)),(,(2

),(2

)()( xthfxhtfhxtfhtxhtx ++++=+

Or equivalently

where

)(21)()( 21 KKtxhtx ++=+

⎧ )(hf

⎩⎨⎧

++==

),(),(

12

1

KxhthfKxthfK

The solution function at t+h is computed at the expenses of two evaluations of the function f

Th t ti f th d d R K tt th d i O(h3)

21

The truncation error of the second‐order Runge‐Kutta method is O(h3)

Page 22: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Compare Euler‘s Method & Runge‐Kutta Methodp g

The initial‐value problem is

1)0(3)(2)(' 4− t

22

1)0( ,3)(2)(' 4 =+−= xetxtx t

Page 23: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Runge‐Kutta Method of Order 4g

The most popular Runge‐Kutta Method is that of order 4,  as

( )1

where

( )432161)()( KKKKtxhtx ++++=+

⎪⎧ =1 ),( xthfK

⎪⎪⎪

⎨⎟⎞

⎜⎛

⎟⎠⎞

⎜⎝⎛ ++= 12

1121,

21

hhf

KxhthfK

( )⎪⎪⎪

⎩ ++=

⎟⎠⎞

⎜⎝⎛ ++=

34

23

,21,

21

KxhthfK

KxhthfK

This formula requires four function evaluations

The truncation error of the fourth‐order Runge‐Kutta method is O(h5)

23

g ( )

Page 24: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Fourth‐Order Runge‐Kutta Methodg

Illustration of Runge‐Kutta method of order 4. Calculations are made at the initial time, two at half of the stepsize beyond the initial time and at the final time These four calculations allow the use of larger overall stepsize with good

24

time. These four calculations allow the use of larger overall stepsize with good accuracy

Page 25: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Fourth‐Order Runge‐Kutta Methodg

The initial‐value problem is 1)0()(' =+= xxytx

25

The initial value problem is 1)0( ,)( + xxytx

Page 26: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Some Amazing Simulation Resultsg

26

Page 27: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Adaptive Runge‐Kutta Methodsp g

In practical computations, we need to know the magnitude of errors i l d i h iinvolved in the computation

Given a tolerance of error, we want to be assured that the computed numerical solution must not deviate from the true solution beyond thenumerical solution must not deviate from the true solution beyond the tolerance

If a method is selected, the error tolerance dictates the allowable step sizeIf a method is selected, the error tolerance dictates the allowable step size

A uniform step size may not be desirable

If the solution is smooth, a large step size may be used to reduce computational cost

27

It is preferable that the method can automatically adjust the step size during the computation

Page 28: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Compute the Errorsp

To advance the solution curve from t to t+h, we can take one step of size hi h f h d R K h dusing the fourth‐order Runge‐Kutta method

We can also take two steps of size h/2 to arrive at t+h

If there were no truncation errors, the values of the numerical solution x(t+h) computed from both procedures would be the same

The difference in the numerical results can be taken as an estimate of the local truncation error

If this error is within the prescribed tolerance, the current step size h is satisfactory. If the difference exceeds the tolerance, the step size is halved 

28

This procedure is easily programmed but rather wasteful of computing time

Page 29: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Fehlberg Method of Order 4g

The formula is12197140825

where

5431 51

41042197

25651408

21625)()( KKKKtxhtx −+++=+

⎧where

⎪⎪⎪⎪⎧

⎟⎠⎞

⎜⎝⎛ ++=

=

12

1

1,1),(

KxhthfK

xthfK

⎪⎪⎪⎪

⎨ ⎟⎠⎞

⎜⎝⎛ +++=

⎟⎠

⎜⎝

213

12

329

323,

83

4,

4

KKxhthfK

f

⎪⎪⎪⎪⎪

⎞⎛

⎟⎠⎞

⎜⎝⎛ +−++=

⎠⎝

3214

845368043921977296

21977200

21971932,

1312 KKKxhthfK

29⎪⎪⎪

⎩⎟⎠⎞

⎜⎝⎛ −+−++= 43215 4104

845513

36808216439, KKKKxhthfK

Page 30: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Fehlberg Method of Order 4, RK45g ,

The Fehlberg method of order 4 requires one more function evaluation than the classical Runge Kutta method it is not very attractivethan the classical Runge‐Kutta method, it is not very attractive

However, with one more function evaluation, as

⎞⎛ 111859354481

We can obtain the fifth‐order Runge‐Kutta method, i.e.,

⎟⎠⎞

⎜⎝⎛ −+−+−+= 543216 40

1141041859

1282535442

278,

21 KKKKKxhthfK

65431 552

509

5643028561

128256656

13516)()( KKKKKtxhtx +−+++=+

The difference between the values of x(t+h) computed from the fourth and fifth order procedures is an estimate of the local truncation error in th f th d d S i f ti l ti i f th

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the fourth‐order procedure. So six function evaluations give a fourth‐order approximation, together with an error estimate

Page 31: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Adaptive Process

1. Given a step size h and an initial value x(t), the RK45routine computes the value x(t+h) and an error estimate  ε

2. If                      , then the step size h is not changed and the next step size is taken by replacing step 1 with initial value x(t+h)

f h h l d b h d d

maxmin εεε ≤≤

≤ |2| hh ≤3. If               , then h is replaced by 2h, provided 

4. If                , then h is replaced by h/2, provided 

minεε ≤ max|2| hh ≤

min|2/| hh ≥maxεε >

5. If                        , then the step size is repeated by returning to step 1 with x(t) and the new h value

maxmin || hhh ≤≤

Here                   are the maximum and minimum error tolerances allowedminmax,εε

hh

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are the maximum and minimum step sizes allowedminmax,hh

Page 32: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Adams‐Bashforth‐Moulton Formulas

Let us consider the first‐order ordinary differential equation

))(,()(' txtftx =If we assume the values of the unknown function have been computed at several points to the left of t, i.e., t, t‐h, t‐2h, …, t‐(n‐1)h, we want to 

( h) h h f l l h

))(,()( txtftx

compute x(t+h). Using the theorem of calculus, we have

)(')()( ∫+

+=+ht

tdssxtxhtx

)(

))(,()(

∫+

+≈

+=n

jj

ht

t

fctx

dssxsftx

where the abbreviation fj is

)( 1

∑=

+j

jj fctx

)))1(()1(( hjthjtff

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)))1((,)1(( hjtxhjtff j −−−−=

Page 33: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Adams‐Bashforth Formulas

The last step needs a rule of the form

∫ +++≈1

)1()1()0()( nFcFcFcdrrF L

We need to determine the n coefficients cj. We can insist on integrating each function                                     exactly. The appropriate equation is

∫ −++−+≈0 21 )1()1()0()( nFcFcFcdrrF nL

12 ,,,,1 −nrrr L

A linear system of the form Ax=b of n equations can be solved for n

∫ ∑=

−− ≤≤−=1

01

11 )1( )1(n

j

ij

i nijcdtr

A linear system of the form Ax=b of n equations can be solved for nunknowns. The coefficients of the matrix A are

1)1( −−= iij jA

and the right‐hand side is  bi =1/i

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Page 34: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Adams‐Moulton Formulas

In Adams‐Moulton formulas, the quadrature is of the form 

Use the change of variable from s to p given by s=hp t the new integral will

∫ ∑=

−≈1

01

)2()(n

jj jGCdrrG

Use the change of variable from s to p given by s=hp‐t, the new integral will be

∫ +1

0)( dpthpgh

We now have

[ ])3(9)2(37)(59)(55)( htFhtFhtFtFhdrrFht

−−−+−−≈∫+ [ ]

[ ])2(249)(5)(19)(924

)(

)3(9)2(37)(59)(5524

)(

htGhtGtGhtGhdrrG

htFhtFhtFtFdrrF

ht

t

t

−−−−++≈

+

∫+

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Page 35: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Stability Analysisy y

For some initial value problems, a small error in the initial value may d f l i d d h d l i iproduce errors of large magnitude, and the computed solution is 

completely wrong

The truncation error in each step further deteriorates the computedThe truncation error in each step further deteriorates the computed solution

For other initial value problems, the situation is not that severeFor other initial value problems, the situation is not that severe

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Page 36: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

A Case of Diverging Solutionsg g

The initial values are close to each other. But over the time, the solutions diverge to different paths 

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Page 37: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

A Case of Converging Solutionsg g

The initial values are quite different. But over the time, the l ti t thi l t h th

37

solutions  converge to something very close to each other

Page 38: Lecture Numerical Solution of Ordinary Differential …jzhang/CS537/lecture8.pdfLecture Numerical Solution of Ordinary Differential Equations Professor Jun Zhang Department of Computer

Solution Stability y

For the general differential equation

⎧ = )xt(f'x

⎩⎨⎧

==

s)a(x)x,t(fx

If

0>> δδ somefor fx

the curves diverge. If

0>< δδ ff

then the curves converge.

0>−< δδ somefor fx

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