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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Euler’s Method. Compute the approximations of y ( t ) at a set of ( usually equally-spaced ) mesh points a = t 0 < t 1 <…< t n = b . That is, to compute w i  y ( t i ) = y i for i = 1, …, n.

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Euler’s Method

Compute the approximations of y(t) at a set of ( usually equally-spaced ) mesh points a = t0< t1 <…< tn= b.

That is, to compute wi y(ti) = yi for i = 1, …, n.

 5.2 Euler’s Method

Theorem: Suppose f is continuous and satisfies a Lipschitz condition with constant L on D = { (t, y) | a t  b, – < y <  } and that a constant M exists with |y”(t)|  M for all a t  b. Let y(t) denote the unique solution to the IVP y’ (t) = f(t, y), a t  b, y(a) = , and w0, w1 , …wn be the approximations generated by Euler’s method for some positive integer n. Then for i = 0, 1, …, n

Difference equation

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Euler’s Method

+ 0

+ i+1

Note: y”(t) can be computed without explicitly knowing y(t).

The roundoff error

Theorem: Let y(t) denote the unique solution to the IVP

y’ (t) = f(t, y), a t  b, y(a) = ,

and w0, w1 , …wn be the approximations obtained using the above difference equations. If | i | <  for i = 0, 1, …, n, then for each i

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Higher Order Taylor Methods

Definition: The difference method

w0 = ; wi+1 = wi + h( ti, wi), for each i = 0, 1, …, n – 1

has local truncation error

for each i = 0, 1, …, n – 1.

 5.3 Higher Order Taylor Methods

Note: The local truncation error is just (yi+1wi+1)/h with the assumption that wi=yi.

The local truncation error of Euler’s method

Method of order 1

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Higher Order Taylor Methods

Taylor method of order n:

where

Note: Euler’s method can be derived by using Taylor’s expansion with n = 1 to approximate y(t).

Higher Order Taylor Methods

The local truncation error is O(hn) if y Cn+1[a, b].

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Higher Order Taylor Methods

T (2)(ti, wi) =

Taylor’s method of order 2:

Example: Apply Taylor’s method of order 2 and 4 to the IVP

y’ = y – t2 + 1, 0  t 2, y(0) = 0.5

Solution: Find the first 3 derivatives of f

f(t, y(t)) = y(t) – t2 + 1

HW:

p.271 #5 (a)(b)

f ’(t, y(t)) = y’(t) – 2t = y(t) – t2 + 1 – 2t

f ”(t, y(t)) = y’(t) – 2t – 2 = y(t) – t2– 2t – 1

f (3)(t, y(t)) = y’(t) – 2t – 2 = y(t) – t2– 2t – 1

Given n = 10, then

h = 0.2 and ti = 0.2i

Table 5.3 on p.269

wi+1 = 1.22wi – 0.0088i2– 0.008i + 0.22

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Higher Order Taylor Methods

+

=

a

+

y

(

t

)

y

(

t

)

h

y’

(

t

)

h

f

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))

1

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=

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w

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+

+

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Hey! Isn’t the local truncation error

of Euler’s method ?

Seems that we can make a good use of it …

Other Euler’s Methods

 Implicit Euler’s method

Usually wi+1 has to be solved iteratively, with an initial value given by the explicit method.

The local truncation error of the implicit Euler’s method

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Higher Order Taylor Methods

 Trapezoidal Method

Two initial points are required to start

moving forward. Such a method is called

double-step method. All the previously discussed

methods are single-step methods.

Note: The local truncation error is indeed O(h2). However an implicit equation has to be solved iteratively.

 Double-step Method

Note: If we assume that wi – 1=yi– 1and wi=yi, the local truncation error is O(h2).

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Higher Order Taylor Methods

Method

Euler’s explicit

Euler’s implicit

Trapezoidal

Double-step

Simple

Low order accuracy

stable

Low order accuracy

and time consuming

More accurate

Time consuming

More accurate,

and explicit

Requires one extra

initial point

Can’t you give me a method

with all the advantages yet without any

Do you think it possible?

Well, call me greedy…

OK, let’s make it possible.

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Higher Order Taylor Methods

Step 1:Predict a solution by the explicit Euler’s method

=

+

w

w

h

f

(

t

,

w

)

+

1

i

i

i

i

Step 2: Correctwi+1 by Plugging it into the right hand side of the trapezoidal formula

h

=

+

+

w

w

[

f

(

t

,

w

)

f

(

t

,

w

)]

+

+

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1

1

1

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2

 Modified Euler’s Method

Note: This kind of method is called the predictor-corrector method. This modified Euler’ method is a single-step method of order 2. It is simpler than the implicit methods and is more stable that the explicit Euler’s method.

9/14

Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Runge-Kutta Methods

A single-step method with high-order local truncation error without evaluating the derivatives of f.

In a single-step method, a line segment is extended from (ti, wi) to reach the next point (ti+1, wi +1) according to some slope. We can improve the result by finding a better slope.

Idea

 5.4 Runge-Kutta Methods

Must the slope

be the average of

K1and K2 ?

Check the modified Euler’s method:

Must the step size be h?

10/14

Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Runge-Kutta Methods

Generalize it to be:

=

+

l

+

l

w

w

h

[

K

K

]

+

1

1

1

2

2

i

i

=

K

f

(

t

,

w

)

1

i

i

=

+

+

K

f

(

t

ph

,

w

phK

)

2

1

i

i

Step 2: Plug K2 into the first equation

Determine 1, 2, and p such that the method has local truncation error of order 2.

Step 1: Write the Taylor expansion of K2at ( ti , yi ) :

Step 3: Find 1, 2, and p such thati+1 = ( yi+1 – wi+1 )/h = O(h2).

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Runge-Kutta Methods

Here are unknowns

and equations.

3

2

There are infinitely many solutions. A family of methods generated from these two equations is called Runge-Kutta method of order 2.

Note: The modified Euler’s method is only a special case of Runge-Kutta methods with p = 1 and 1 = 2 = 1/2.

Q: How to obtain higher-ordered accuracy?

12/14

Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Runge-Kutta Methods

=

+

l

+

l

+

+

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w

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[

K

K

...

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1

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-

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1

m

1

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2

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m

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m

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m

m m

Classical Runge-Kutta Order 4 Method – the most popular one

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Chapter 5 Initial-Value Problems for Ordinary Differential Equations -- Runge-Kutta Methods

Note:

The main computational effort in applying the Runge-Kutta methods is the evaluation of f. Butcher has established the relationship between the number of evaluations per step and the order of the local truncation error :

evaluations per step

2

3

4

5-7

8-9

n10

Best possible LTE

HW:

p.280-281

#1 (a), 10, 13

Since Runge-Kutta methods are based on Taylor’s expansion, y has to be sufficiently smooth to obtain better accuracy with higher-order methods. Usually the methods of lower order are used with smaller step size in preference to the higher-order methods using a large step size.

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