1. Class Notes 18: Numerical Methods (1/2)
82 – Engineering Mathematics

2. Differential Equation
Numerical Methods
Differential Equation
Analytical Techniques:
Integrating Factor & Exact
(1st order)
(2nd or higher order, linear, constant coefficients)
Power Series
(2nd order, linear)
Laplace Transform
(Linear, constant coefficient, I.C.)
Numerical Techniques:
Fix (single) step
- Euler
- Runge-Kutta
Multi-step
- Adams-Bashforth
Idea: approximate the value of y at a specified time tn

3. Numerical Methods: the Euler Method
WHAT is Euler Method
WHY Euler method can work
HOW to apply Euler method
What is the PROBLEM with Euler method
How to IMPROVE Euler method

4. The Euler Method (Tangent Line Method) - WHAT
First order initial value problem
Euler’s Formula:
Repeatedly evaluating Euler’s formula using the result of each step to execute the next step
Obtain a sequence of values y0, y1, y2, … yn that approximate the value of the solution at points t0, t1, t2, … tn
Euler’s formula can be derived in three ways

5. The Euler Method (Tangent Line Method) - WHY
Approach 1: Tangent Line
Assume an unique solution of the form
Write the differential equation at the point t=tn
Approximate the derivation by the corresponding forward difference quotient

6. The Euler Method (Tangent Line Method) - WHY
Replace
Solve for
h → step size

7. The Euler Method (Tangent Line Method) - WHY
Approach 2: Integration
Let be the solution of the initial value problem

8. The Euler Method (Tangent Line Method) - WHY
Replace
shaded rectangle

9. The Euler Method (Tangent Line Method) - WHY
Approach 3: Taylor Series
The solution has a Taylor series about tn or
replace and take the first two terms (linearization)

10. The Euler Method (Tangent Line Method) - HOW
How to apply Euler method
step 1 define f(t,y)
step 2 input initial values t0 and y0
step 3 input step size h and the number of steps n
step 4 output t0 and y0
step 5 for j from 1 to n do
k1 = f(t, y)
y = y + h*k1 n times
t = t + h
output t and y
end

11. The Euler Method (Tangent Line Method) - HOW
Example
Exact solution

12. The Euler Method (Tangent Line Method) - PROBLEM
Error in numerical approximations
Convergence
How small a step size is needed in order to guarantee a given level of accuracy?
slow down calculations
Small step size -
may cause loss of accuracy
Numerical approximation
Analytical solution
Step size

13. The Euler Method (Tangent Line Method) - PROBLEM
Fundamental sources of errors
1) Formula/Algorithm - approximation
e.g. Euler → straight line approximations
2) The input data (except for the first step) are only approximations to the actual values of the solution at the specific points
3) Finite precision of the computer

14. The Euler Method (Tangent Line Method) - PROBLEM
Error type 1) and 2)
Assume that the computer can execute all computations exactly. It can retain infinitely many digits (if necessary) at each step
Error type 3)
Global Truncation Error
(source of error: 1 and 2)
Solution
Numerical Approximation
Actual computed solution
With round off error
Round off error
Numerical Approximation

15. The Euler Method (Tangent Line Method) - PROBLEM
The absolute value of the total error in computing
based on
Global Truncation Error
Round off error

16. The Euler Method (Tangent Line Method) - PROBLEM
Example – Round off Error
Computer with four digits
Compute for x=0.3334

17. The Euler Method (Tangent Line Method) - PROBLEM
How to reduce round-off error
1) Minimize the number of calculations
2) Use double-precision arithmetic

18. The Euler Method (Tangent Line Method) - PROBLEM
Global versus Local Truncation Error
Assume that the solution is
to the initial value problem
Global Truncation Error
Local Truncation Error
(In each step)

19. The Euler Method (Tangent Line Method) - PROBLEM
Expand about tn using polynomial with a remainder
where is some point
using the Euler formula to calculate an approximation to
the difference between and is the local truncation error

20. The Euler Method (Tangent Line Method) - PROBLEM
M is the maximum of on the interval [a b]

21. The Euler Method (Tangent Line Method) - PROBLEM
One use of the equation is to choose a step size that will result in a local truncation error no greater than some given tolerance level
For example if the local truncation error must be no greater than ε
Difficulty: estimate or M

22. The Euler Method (Tangent Line Method) - PROBLEM
Reducing the interval by ½ reduces the error by ¼
The global truncation error
Euler method is called a first order method because its global truncation error is proportional to the first power of the step size h
K: some constant

23. The Euler Method (Tangent Line Method) - IMPROVE
Improved Euler formula (Heun formula)
Idea: replace integrand by the average of the two endpoints
Heun formula
Euler formula

24. The Euler Method (Tangent Line Method) - IMPROVE
Improved Euler formula (Heun formula)
Heun formula
Euler formula
Local Truncation Error:
Local Truncation Error:
Global truncation error: First order method
Global Truncation Error: Second order method
Cost: more computational work

25. The Euler Method (Tangent Line Method) - IMPROVE
If depends only on t and not on y, solving the differential equation reduces to integrating in this case
Trapezoid rule of numerical integration

26. The Euler Method (Tangent Line Method) - IMPROVE
How to apply improved Euler method
step 1 define f(t,y)
step 2 input initial values t0 and y0
step 3 input step size h and the number of steps n
step 4 output t0 and y0
step 5 for j from 1 to n do
k1 = f(t, y)
k2 = f(t + h, y + h*k1)
y = y (h/2)*(k1 + k2)
t = t + h
output t and y
end

27. The Euler Method (Tangent Line Method) - IMPROVE
The improved Euler method is an example of a predictor-corrector method
predicts the value of yn+1
corrects the estimate
(Predictor)
(Corrector)