Review

Initial Value Problems

Stability
Summary

E ULERS M ETHOD AND S TABILITY

Dr. Johnson

School of Mathematics

Semester 1 2012

Dr. Johnson MATH49111

 Review
Initial Value Problems
Stability
Summary

L AST W EEK ...

Topics:
Errors and precision
Flow control - if, else, for, do
Functions - how and where to use them
Pointers - dynamic allocation, pass by reference
Aims - week 2:
Understand precision and errors within a program.
Use functions, loops and if statement to control a program.
Understand the concept of a pointer.

Dr. Johnson MATH49111

 Review
Initial Value Problems
Stability
Summary

T HIS W EEK ...

Topics:
Initial value problems for ODEs
Euler’s Method for ODEs
Stability
Using the standard libraries
Aims - week 3:
Be able to solve an initial value ODE problem
Understand where truncation errors come from
Be aware of the concept of stability in a numerical method
Be able to use standard libraries to store your data

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

T HE I NITIAL VALUE P ROBLEM

Here we will look at the solution of ordinary differential

equations of the type, say

dy
= f (x, y ), a≤x ≤b
dx

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

T HE I NITIAL VALUE P ROBLEM

Here we will look at the solution of ordinary differential

equations of the type, say

dy
= f (x, y ), a≤x ≤b
dx
subject to an initial condition

y (a) = α

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

E XAMPLES

C ONSIDER THE PROBLEM :

dy  y
=y 1− , x ≥0
dx 4
with the initial condition
y (0) = 1

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

E XAMPLES

C ONSIDER THE PROBLEM :

dy  y
=y 1− , x ≥0
dx 4
with the initial condition
y (0) = 1

How can we go about solving it?

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

S OLVING AN I NITIAL VALUE P ROBLEM

The simplest method to solve an ODE is the Euler method

In order to solve, we must discretise the problem – make a
continuous (infinite) problem discrete (finite)
Divide up the interval [a, b] into n equally spaced intervals
h
a b

x0 x1 x2 ... xi ... xn

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

S OLVING AN I NITIAL VALUE P ROBLEM

Suppose y (x) is the unique solution to the ODE, and is

twice differentiable
Apply a Taylor series approximation around xi then we
have

1
y (xi+1 ) = y (xi ) + y 0 (xi )h + y 00 (ξ)
2
where xi ≤ ξ ≤ xi+1 .

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

S OLVING AN I NITIAL VALUE P ROBLEM

Suppose y (x) is the unique solution to the ODE, and is

twice differentiable
Apply a Taylor series approximation around xi then we
have

1
y (xi+1 ) = y (xi ) + f (xi , y (xi ))h + y 00 (ξ)
2
where xi ≤ ξ ≤ xi+1 .

Since y is a solution to the ODE, we can replace y 0 by the

function f (x, y )
Dr. Johnson MATH49111
 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

T HE E ULER M ETHOD

Assume wi is our approximation to y at xi , then

w0 = α

Then to find all subsequent values of w,set the remainder

to zero in the previous equation to obtain:

wi+1 = wi + hf (xi , wi ), for each i = 0, 1, . . . , n − 1

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

T HE E ULER M ETHOD

Assume wi is our approximation to y at xi , then

w0 = α

Then to find all subsequent values of w,set the remainder

to zero in the previous equation to obtain:

wi+1 = wi + hf (xi , wi ), for each i = 0, 1, . . . , n − 1

Try solving the example problem...

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

T RUNCATION E RRORS

We would like to be able to compare the errors for different

methods
We can use the local truncation error – difference between
equation and the approximation

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

T RUNCATION E RRORS

We would like to be able to compare the errors for different

methods
We can use the local truncation error – difference between
equation and the approximation
For the Euler method we have:

yi+1 − (yi + hf (xi , yi ))

τi+1 (h) =
h
yi+1 − yi
= − f (xi , yi )
h

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

T RUNCATION E RRORS

We can calculate the truncation error as

h 00
τi+1 (h) = y (ξi ),
2

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

T RUNCATION E RRORS

We can calculate the truncation error as

h 00
τi+1 (h) = y (ξi ),
2
and if y 00 is bounded by the constant M on the interval
[a, b] then

h
|τi+1 (h)| ≤ M.
2

Dr. Johnson MATH49111

 Review
ODE Problems
Initial Value Problems
Euler’s Method
Stability
Errors
Summary

T RUNCATION E RRORS

We can calculate the truncation error as

h 00
τi+1 (h) = y (ξi ),
2
and if y 00 is bounded by the constant M on the interval
[a, b] then

h
|τi+1 (h)| ≤ M.
2
Hence the truncation error is O(h).
A method with truncation error O(hp ) is called an order p
method
Dr. Johnson MATH49111
 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

A N EXAMPLE PROBLEM :

Suppose the initial-value problem

y 0 = −50y , 0 ≤ x ≤ 1, y (0) = 1

is approximated by the Euler method so that

yi+1 = yi + hf (xi , yi ).

The exact solution to the problem is

y (x) = e−50x

This is a stiff problem.

Dr. Johnson MATH49111

 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

T HE E XACT S OLUTION

1
y=e-50x

0.8

0.6
y(x)

0.4

0.2

0
0 0.2 0.4 0.6 0.8 1
x

Dr. Johnson MATH49111

 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

U NSTABLE S OLUTION

Now we generate the numerical solution with h = 0.05, we

have that
y0 = 1
y1 = 1 + (0.05) · (−50 · 1) = −1.5
y2 = −1.5 + (0.05) · (−50 · −1.5) = 2.25
and so on. . .
Here errors are growing, the method is unstable.

Dr. Johnson MATH49111

 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

U NSTABLE SOLUTION

Now we take h = 0.04, then

y0 = 1
y1 = 1 + (0.04) · (−50 · 1) = −1
y2 = −1 + (0.04) · (−50 · −1) = 1
and so on. . .
Here the solution oscilates, and errors grow slowly. The
method is unstable.

Dr. Johnson MATH49111

 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

S TABLE SOLUTION

Next take h = 1/30, then

y0 = 1
y1 = 1 + (1/30) · (−50 · 1) = −2/3
y2 = −2/3 + (1/30) · (−50 · −2/3) = 4/9
and so on. . .
Here the solution oscilates, but errors reduce slowly. The
method is in some sense stable.

Dr. Johnson MATH49111

 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

U NSTABLE S OLUTIONS

y=e-50x
4 n=20
n=25
n=30

2
y(x)

-2

-4

0 0.2 0.4 0.6 0.8 1

x

Dr. Johnson MATH49111

 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

S TABLE S OLUTIONS

A stable solution is one in which errors or pertubations are

damped down.
Think of the first case with h = 0.05:
The error from calculating y1 is amplified at y2 .
Given the exact value for y1 , the error at y2 would be much
smaller.
If we set h small enough the errors are bounded and the
method gives satisfactory results.

Dr. Johnson MATH49111

 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

S OME S TABLE S OLUTIONS

1
y=e-50x
n=50
n=100
0.8

0.6
y(x)

0.4

0.2

0
0 0.2 0.4 0.6 0.8 1
x

Dr. Johnson MATH49111

 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

C ONSISTENCY

D EFINITION :
A one-step difference method with truncation error τi (h) at the
ith step is said to be consistent with the difference equation it
approximates if
lim max |τi (h)| = 0.
h→0 i

This is a local definition.

We compare the exact value to the difference
approximation given that we know the exact value at yi−1 .

Dr. Johnson MATH49111

 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

C ONVERGENCE

D EFINITION :
A one-step difference method is said to be convergent with
respect to the difference equation it approximates if

lim max |wi − y (xi )| = 0,

h→0 i

where wi is the approximate value at xi to the exact value

yi = y (xi ).

This is a global definition.

Given that only y0 is exact, the errors at the ith position
tend to zero as we reduce the step size.

Dr. Johnson MATH49111

 Review
Initial Value Problems What is stability?
Stability Defining stability
Summary

S TABILITY

D EFINITION :
A stable method is one whose results depend continuously on
the initial data.

Consider a one step method in the form

wi+1 = wi + hφ(xi , yi , h),

such that φ is continuous and satisfies a Lipschitz

condition. Then the method is stable as defined above.
If the method is consistent, then it is convergent if and only
if it is stable.

Dr. Johnson MATH49111

 Review
Initial Value Problems
Stability
Summary

T HIS W EEK ...

Topics:
Initial value problems for ODEs
Euler’s Method for ODEs
Stability
Using the standard libraries
Aims - week 3:
Be able to solve an initial value ODE problem
Understand where truncation errors come from
Be aware of the concept of stability in a numerical method
Be able to use standard libraries to store your data

Dr. Johnson MATH49111