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Lecture 21 - Ordinary Differential Equations -

IVP

- CVEN 302
- July 20, 2001

Lectures Goals

ODE Methods

- 4th order Runge-Kutta Method
- Multi-step Method
- Adam Bashforth
- Adam Moulton Method
- Predictor-Corrector Method
- Stability

Runge-Kutta Methods

The Runge-Kutta methods are higher order

approximation of the basic forward integration.

These methods provide solutions which are

comparable in accuracy to Taylor series solution

in which higher order derivatives are retained.

It should be noted that the equations are not

need to be linear.

Runge-Kutta Methods

The 4th order Runge-Kutta

This is a fourth order function that solves an

initial value problems using a four step program

to get an estimate of the Taylor series through

the fourth order. This will result in a local

error of O(Dh5) and a global error of O(Dh4)

The 4th order Runge-Kutta

The general form of the equations

4th-orderRunge-Kutta Method

f2

f4

f3

f1

xi

xi h/2

xi h

Runge-Kutta Method (4th Order) Example

Consider Exact

Solution The initial condition is The step

size is

The 4th order Runge-Kutta

The example of a single step

Runge-Kutta Method (4th Order) Example

The values for the 4th order Runge-Kutta method

Runge-Kutta Method (4th Order) Example

The values are equivalent to those of the exact

solution. If we were to go out to x5. y(5)

-111.4129 (-111.4132) The error is small relative

to the exact solution.

Runge-Kutta Method (4th Order) Example

A comparison between the 2nd order and the 4th

order Runge-Kutta methods show a slight

difference.

The 4th order Runge-Kutta

Higher order differential equations can be

treated as if they were a set of first-order

equations. Runge-Kutta type forward integration

solutions can be obtain. A more direct solution

can be obtained by repeating the whole process

used in first-order cases.

The 4th order Runge-Kutta

The general form of the equations

The 4th order Runge-Kutta

The step sizes are

The next step would be

One Step Method

Up until this point we have dealt with These

methods are called single step methods, because

they use only the information from the previous

step.

- Euler Method
- Modified Euler/Midpoint
- Runge-Kutta Methods

One Step Method

The techniques are defined as

- These methods allow us to vary the step size.
- Use only one initial value
- After each step is completed the past step is

forgotten We do not use this information.

Multi-Step Methods

The principle behind a multi-step method is to

use past values, y and/or dy/dx to construct a

polynomial that approximate the derivative

function.

Multi-Step Methods

The method comes from integrating the functions.

Multi-Step Methods

The integral can be represented.

Multi-Step Methods

The integral can be represented.

Multi-Step Methods

These methods are known as explicit schemes

because the use of current and past values are

used to obtain the future step. The method is

initiated by using either a set of know results

or from the results of a Runge-Kutta to start the

initial value problem.

Adam Bashforth Method (4th Point) Example

Consider Exact

Solution The initial condition is The step

size is

4 Point Adam Bashforth

From the 4th order Runge Kutta The 4 Point

Adam Bashforth is

4 Point Adam Bashforth

The results are Upgrade the values

4 Point Adam Bashforth Method - Example

The values for the Adam Bashforth

4 Point Adam Bashforth Method - Example

The explicit Adam Bashforth method gave solution

gives good results without having to go through

large number of calculations.

Multi-Step Methods

There are second set of multi-step methods, which

are known as implicit methods. The implicit

methods use the future steps to modify the future

steps. What is used to do iterative method,

which will make an initial guess and use it until

stability is reached. The method is initiated by

using either a set of know results or from the

results of a Runge-Kutta to start the initial

value problem.

Implicit Multi-Step Methods

The main method is Adam Moulton Method

Implicit Multi-Step Methods

The method uses what is known as a

Predictor-Corrector technique. It uses the

explicit scheme to estimate the initial guess and

uses the value to guess the future y and dy/dx

f(x,y) values. Using these results, the Adam

Moulton method can be applied.

Implicit Multi-Step Methods

Adams third order Predictor-Corrector

scheme. Use the Adam Bashforth three point

explicit scheme for the initial guess. Use

the Adam Moulton three point implicit scheme to

take a second step.

Adam Moulton Method (3th Point) Example

Consider Exact

Solution The initial condition is The step

size is

4 Point Adam Bashforth

From the 4th order Runge Kutta The 3 Point

Adam Bashforth is

3 Point Adam Moulton Predictor-Corrector Method

The results of explicit scheme is The

functional values are

3 Point Adam Moulton Predictor-Corrector Method

The results of implicit scheme is The

functional values are

3 Point Adam Moulton Predictor-Corrector Method

The values for the Adam Moulton

3 Point Adam Moulton Predictor-Corrector Method

The implicit Adam Moulton method gave solution

gives good results without using more than a

three points.

Numerical Stability

- Amplification or decay of numerical errors
- A numerical method is stable if error incurred at

one stage of the process do not tend to magnify

at later stages - Ill-conditioned differential equation
- -- numerical errors will be magnified

regardless - of numerical method
- Stiff differential equation
- -- require extremely small step size to

achieve - accurate results

Stability

- Example problem

Explicit Euler Method

- Stability criterion
- Region of absolute stability

Stability

- Explicit Euler method
- Second-order Adams-Bashforth
- Second-order Adams-Moulton

Summary

- 4th order Runge-Kutta Method
- Higher order Runge-Kutta Methods
- Explicit Multi-Step Methods
- Implicit Multi-Step Methods
- Stability

Homework

- Check the Homework webpage

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