numerical method heun's method
DESCRIPTION
heun methodmid point methodeuler methodTRANSCRIPT
-
Lecture 2: Introduction to Ordinary Differential Equation (ODE) and Euler Method KKEK214:2 Numerical Methods for Engineers 2
Department of Chemical Engineering
-
Differential Equation
Definition: a mathematical equation that consists of an unknown function and its derivatives.
It expresses the relation between a function and its derivatives.
Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
Examples: reaction rate, mass flow, heat transfer rate, acceleration, velocity, etc.
vm
cg
dt
dv
MDH-KKEK2142 (Sem 2 2014/2015)
2
E.g.: Newtons Second Law (1)
Where: = velocity g = gravitational constant t = time m = mass c = drag coefficient
- dependent variable
t- independent variable
-
Differential Equation
When a function involves one independent variable, the equation is called an ordinary differential equation (or ODE).
A partial differential equation (or PDE) involves two or more independent variables.
MDH-KKEK2142 (Sem 2 2014/2015) 3
-
4 MDH-KKEK2142 (Sem 2 2014/2015)
Example: ODE in chemical reaction problem
A chemical reaction of the type of
(2)
takes place in a reactor, the material balance can be applied as:
Input = Output +
+ Generation Accumulation
(3)
For a batch reactor,
Input = Output +
+ Generation Accumulation
CBAk
k
1
2
Hence,
ODE
CBAC
CBAB
CBAA
CkCCkdt
dC
CkCCkdt
dC
CkCCkdt
dC
21
21
21
-
Classification of ODEs
ODEs are classified according to their order and their linearity.
The order of an ODE is the order of the highest derivative present in that equation.
1st order
2nd order
3rd order kx
dx
dyb
dx
yda
dx
yd
kxdx
dyy
dx
yd
kxydx
dy
2
2
2
3
3
2
2
MDH-KKEK2142 (Sem 2 2014/2015) 5
-
Classification of ODEs
6 MDH-KKEK2142 (Sem 2 2014/2015)
ODEs also can be classified according to its linearity, i.e. linear and nonlinear ODE.
An ODE is nonlinear if it contains product of the dependent variable or its derivatives or both.
Linear
Nonlinear
Nonlinear
2
2
23 2
3 2
dyy kx
dx
d y dyy kx
dx dx
d y d y dya b kx
dx dx dx
-
Degree of ODEs
The degree of an ODE is the largest power to which the highest order derivative is raised.
Has degree of 5
Has degree of 2
kxdx
dyb
dx
yda
dx
yd
kxdx
dyb
dx
yda
dx
yd
105
2
22
3
3
2
2
25
3
3
MDH-KKEK2142 (Sem 2 2014/2015)
7
-
Runge-Kutta Methods
However not all ODEs can be solved using analytical methods of calculus.
Need to use numerical methods.
Methods used to solve ODEs using iterative methods.
Developed by German mathematicians C. Runge and M.W. Kutta.
),( yxf
dx
dy
MDH-KKEK2142 (Sem 2 2014/2015) 8
-
Runge-Kutta Methods
Eulers method
Heuns method
Midpoint method
Higher Order Runge-Kutta (RK) method
9 MDH-KKEK2142 (Sem 2 2014/2015)
-
Runge-Kutta Methods
Solve ordinary differential equations of the form
Numerical method in the general form of
= +
or in mathematical terms
+1 = +
Applied step by step to compute out the value in the future (trajectory of the solution).
10 MDH-KKEK2142 (Sem 2 2014/2015)
),( yxfdx
dy
-
Runge-Kutta Methods
The type of RK methods can be differentiated by the manner in which the slope is estimated.
The easiest one-step method: Eulers method.
However, Eulers method has the least accurate predictions compared to other RK methods.
11 MDH-KKEK2142 (Sem 2 2014/2015)
-
Eulers Method
Named after Leonhard Euler (1707-1783).
Swiss mathematician and physicist.
He introduced several mathematical notations such as f(x), e (also known as Eulers number),
for summation and i.
He also had introduce the Eulers method to solve ordinary differential equation.
It is the simplest method to solve an ODE with a given initial value, yo.
MDH-KKEK2142 (Sem 2 2014/2015) 12
-
Eulers Method
The first derivative provides a direct estimate of the slope at xi
where f(xi,yi) is the differential equation evaluated at xi and yi. This estimate can be substituted into the equation:
A new value of y is predicted using the slope to extrapolate linearly over the step size h.
MDH-KKEK2142 (Sem 2 2014/2015) 13
Figure 25.2 Eulers method
),( ii yxf
hyxfyy iiii ),(1
-
14 MDH-KKEK2142 (Sem 2 2014/2015)
Example (pg. 708 in Text Book):
Use Eulers method to numerically integrate
From x=0 to x=4 with a step size of 0.5. The initial
condition at x=0 is y=1.
5.820122 23 xxxdx
dy
-
Exercise (homework)
Calculate the values from previous example for the subsequent steps, i.e. y(1.0), y(1.5), y(2.0), y(2.5), y(3.0), y(3.5) and y(4.0).
Compare the value that you obtained using Eulers method with those obtained using exact solution of
15 MDH-KKEK2142 (Sem 2 2014/2015)
= 0.54 + 43 102 + 8.5 + 1
-
Error Analysis of the RK Method
Numerical solutions of ODEs involves two types of error:
i) Truncation/discretisation errors
Caused by the nature of the techniques employed to approximate the value of y.
ii) Round off errors
Caused by the limited numbers of significant digits
16 MDH-KKEK2142 (Sem 2 2014/2015)
-
Error Analysis of the RK Method
Truncation errors:
i) Local truncation error that results from an application of the method in question over single step.
ii) Propagated truncation error that results from the approximation produced during the previous steps.
17 MDH-KKEK2142 (Sem 2 2014/2015)
-
Error Analysis of the Eulers Method (Cont.)
Total error = Local truncation error + Propagated truncation error
18 MDH-KKEK2142 (Sem 2 2014/2015)
Global truncation error
-
Error Analysis of the RK Method (Cont.)
Conclusions:
i) The global error can be reduced by decreasing h.
ii) The method will provide error-free predictions if the underlying function is linear.
19 MDH-KKEK2142 (Sem 2 2014/2015)
-
MATLAB
Short for MATrix LABoratory
Optimized to perform engineering and scientific calculations.
Initially designed to perform matrix mathematics
Provides a very extensive library of predefined functions to make technical programming tasks easier and more efficient.
20 MDH-KKEK2142 (Sem 2 2014/2015)
Advantages:
1. Ease of use
2. Platform independence
3. Pre-defined functions
4. Device-independent plotting
5. Graphical User Interface
6. MATLAB Compiler
Disadvantages:
1. Interpreted language execute more slowly
2. Cost
-
MATLAB Environment: MATLAB Desktop
21 MDH-KKEK2142 (Sem 2 2014/2015)
The Command Window: User can enter interactive commands at the command prompt (>>) and it will be executed on the spot. It is easier to use script files / M-files
The Command History Window
Displays the commands that user has entered
The MATLAB Workspace
Stores the variable name and value
-
Getting Help
Help Browser
Type help on the command window
Using lookfor command
Other Important Functions
Type demo to run MATLABs built-in demonstrations.
clc : clear the Command Windows content.
clear : clear the variables in the Workspace
abort command by typing control-c
22 MDH-KKEK2142 (Sem 2 2014/2015)
-
Important Programming Pitfalls
Never use a variable with the same name as a MATLAB function or command. If you do so, that function or command will become inaccessible.
If there is more than one function or command with the same name, the first one found on the search path will be executed.
Never create an M-file with the same name as a MATLAB function or command.
23 MDH-KKEK2142 (Sem 2 2014/2015)