The Euler Method - School of Mathematicspjohnson/CFD/Lectures/euler_method… · Review Initial Value Problems Euler’s Method Summary THE EULER METHOD P.V. Johnson School of Mathematics

Review Initial Value Problems Euler’s Method Summary

THE EULER METHOD

P.V. Johnson

School of Mathematics

Semester 1 2008


OUTLINE

1 REVIEW

2 INITIAL VALUE PROBLEMS

The ProblemPosing a Problem

3 EULER’S METHOD

MethodErrors

4 SUMMARY


OUTLINE

1 REVIEW



3 EULER’S METHOD

MethodErrors

4 SUMMARY


OUTLINE

1 REVIEW



3 EULER’S METHOD

MethodErrors

4 SUMMARY


OUTLINE

1 REVIEW



3 EULER’S METHOD

MethodErrors

4 SUMMARY


ERRORS

Be careful when interpreting results from a computer

Errors can arise in all sorts of places

Using Taylor Series to approximate a differential

Truncation errors


ERRORS

Be careful when interpreting results from a computer

Errors can arise in all sorts of places

Using Taylor Series to approximate a differential

Truncation errors


EULER’S METHOD

Initial value problems for ODEs

Solving ODEs numerically

Euler’s Method

Truncation errors again


OUTLINE

1 REVIEW



3 EULER’S METHOD

MethodErrors

4 SUMMARY


THE INITIAL VALUE PROBLEM

Here we will look at the solution of ordinary differentialequations of the type, say

dydx

= f (x , y), a ≤ x ≤ b

subject to an initial condition

y(a) = α


THE INITIAL VALUE PROBLEM

Here we will look at the solution of ordinary differentialequations of the type, say

dydx

= f (x , y), a ≤ x ≤ b

subject to an initial condition

y(a) = α


EXAMPLES

CONSIDER THE PROBLEM:

dydx

= y(

1 −y4

)

, x ≥ 0

with the initial condition

y(0) = 1

What can we say about the solution?

How can we go about solving it?


EXAMPLES

CONSIDER THE PROBLEM:

dydx

= y(

1 −y4

)

, x ≥ 0

with the initial condition

y(0) = 1

What can we say about the solution?

How can we go about solving it?


EXTENSION TO HIGHER ORDER ODES

We normally would like to solve higher order ODEs

We can do this by rearranging them as a system of firstorder ODEs

EXAMPLE

y ′′ − 2xyy ′ + y2 = 1, y(1) = 1, y ′(1) = 2.

The equivalent first order system is:

(y1(x), y2(x))T = (y(x), y ′(x))T ,f1(x , y1, y2) = y2(x),

f2(x , y1, y2) = 1 + 2xy1(x)y2(x) − y21 (x),

and initial condition

(y1(1), y2(1))T = (1, 2)T .





EXAMPLE

y ′′ − 2xyy ′ + y2 = 1, y(1) = 1, y ′(1) = 2.



f2(x , y1, y2) = 1 + 2xy1(x)y2(x) − y21 (x),


(y1(1), y2(1))T = (1, 2)T .





EXAMPLE

y ′′ − 2xyy ′ + y2 = 1, y(1) = 1, y ′(1) = 2.



f2(x , y1, y2) = 1 + 2xy1(x)y2(x) − y21 (x),


(y1(1), y2(1))T = (1, 2)T .


GENERAL SYSTEM OF FIRST ORDER ODES

The general system can be written

dYdx

= F (x , Y ), a ≤ x ≤ b,

where

Y = (y1(x), y2(x), ..., yn(x))T ,F = (f1(x , Y ), f2(x , Y ), ..., fn(x , Y ))T ,

with initial data

Y (a) = α,α = (α1, α2, ..., αn)

T .


OUTLINE

1 REVIEW



3 EULER’S METHOD

MethodErrors

4 SUMMARY


WELL-POSED PROBLEMS

DEFINITION:

An initial value problem is said to be well-posed if:

A unique solution, y(x), to the problem exists;

The associated perturbed problem has a unique solutionand is sufficiently close to the original solution

How do we go about showing this for a general problem?

What do we mean by a perturbed problem?

With a computer, we are always solving the perturbedproblem

First, we need to know about the Lipschitz condition


WELL-POSED PROBLEMS

DEFINITION:









WELL-POSED PROBLEMS

DEFINITION:









THE LIPSCHITZ CONDITION

DEFINITION:

A function is said to satisfy the Lipschitz condition in thevariable y on a set D ⊂ R2 if a constant L > 0 exists such that:

|f (x , y1) − f (x , y2)| ≤ L|y1 − y2|

whenever (x , y1), (x , y2) ∈ D. The constant L is called theLipschitz constant for f .


PERTURBED PROBLEM

THE PERTURBED PROBLEM

Let δ(x) be a pertubation to the problem so that δ(x) < ǫ for anyǫ > 0, then

dzdx

= f (x , z) + δ(x), a ≤ x ≤ b, z(a) = α + ǫ0

and the solution z(x) exists with

|z(x) − y(x)| < k(ǫ)ǫ for all a ≤ x ≤ b,

for some constant k(ǫ).


THE THEORY

THEOREM

Suppose D = {(x , y)|a ≤ x ≤ b and −∞ < y < ∞}. If f iscontinuous and satisfies a Lipschitz condition in the variable yon the set D, then the initial-value problem

dydx

= f (x , y), a ≤ x ≤ b, y(a) = α

is well-posed.


OUTLINE

1 REVIEW



3 EULER’S METHOD

MethodErrors

4 SUMMARY


SOLVING AN INITIAL VALUE PROBLEM

The simplest method to solve an ODE is the Euler method

In order to solve, we must discretise the problem – make acontinuous (infinite) problem discrete (finite)

Divide up the interval [a, b] into n equally spaced intervals

bah

x x x xx0 1 2 i n... ...



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

Apply a Taylor series approximation around xi then wehave

y(xi+1) = y(xi) + y ′(xi)h +12

y ′′(ξ)

where xi ≤ ξ ≤ xi+1 .



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

Apply a Taylor series approximation around xi then wehave

y(xi+1) = y(xi) + f (xi , y(xi))h +12

y ′′(ξ)

where xi ≤ ξ ≤ xi+1 .

Since y is a solution to the ODE, we can replace y ′ by thefunction f (x , y)


THE EULER METHOD

Assume wi is our approximation to y at xi , then

w0 = α

Then to find all subsequent values of w ,set the remainderto zero in the previous equation to obtain:

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


OUTLINE

1 REVIEW



3 EULER’S METHOD

MethodErrors

4 SUMMARY


TRUNCATION ERRORS

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

We can use the local truncation error – difference betweenequation and the approximation

For the Euler method we have:

τi+1(h) =yi+1 − (yi + hf (xi , yi))

h

=yi+1 − yi

h− f (xi , yi)


TRUNCATION ERRORS

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

We can use the local truncation error – difference betweenequation and the approximation

For the Euler method we have:

τi+1(h) =yi+1 − (yi + hf (xi , yi))

h

=yi+1 − yi

h− f (xi , yi)


TRUNCATION ERRORS

We can calculate the truncation error as

τi+1(h) =h2

y ′′(ξi),

and if y ′′ is bounded by the constant M on the interval[a, b] then

|τi+1(h)| ≤h2

M.

Hence the truncation error is O(h).

A method with truncation error O(hp) is called an order pmethod


TRUNCATION ERRORS


τi+1(h) =h2

y ′′(ξi),


|τi+1(h)| ≤h2

M.




TRUNCATION ERRORS


τi+1(h) =h2

y ′′(ξi),


|τi+1(h)| ≤h2

M.




SUMMARY

Need Lipschitz condition to say a problem is well PosedProblems

Euler’s Method is a first order method

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

Truncation errors give a quick estimate of error bounds onthe problem

Documents

The Euler Method - School of Mathematicspjohnson/CFD/Lectures/euler_method… · Review Initial Value Problems Euler’s Method Summary THE EULER METHOD P.V. Johnson School of Mathematics