ChE 250 Numeric Methods

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ChE 250 Numeric Methods. Lecture #20 Part 6 Introduction Chapra, Chapter 21: Newton-Cotes Chapter 22: Integration of Equations 20070326. Class Schedule. Two weeks of lecture left This week, Chapter 21-23 Next week, Chapter 24-25 We need to schedule your final exam

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### ChE 250 Numeric Methods

Lecture #20

Part 6 Introduction

Chapra, Chapter 21: Newton-Cotes

Chapter 22: Integration of Equations

20070326

Class Schedule
• Two weeks of lecture left
• This week, Chapter 21-23
• Next week, Chapter 24-25
• We need to schedule your final exam
• Cumulative chapters 1 through 22
• Choices: 11th, 13th,…other?
• Project 2
• Due April 20th
• Should be posted on course webpage next weekend
Today’s Lecture
• Introduction
• What integration and differentiation is used for
• Chapter 21: Newton-Cotes Integration
• Trapezoid Rule
• Simpson’s 1/3 Rule
• Simpson’s 3/8 Rule
• Composite Methods
• Chapter 22: Integration of Equations
• Romberg Integration
Numeric Integration & Differentiation
• Many times, it is not efficient to analytically derive the solution to an integral problem so numeric methods are used
• If analyzing a data set, two approaches are available:
• you can use interval methods like the trapezoidal rule or Simpson’s
• you can use techniques discussed in part 5 to fit the data to a curve, then use calculus to calculate the definite integral
Numeric Integration & Differentiation
• If the derivative is needed there are finite difference methods that can be used
• Finite difference works for data or functions
• it is preferred to use a curve instead of raw so that error does not propagate
Newton-Cotes Integration Formulas
• For data or a function, the easy way to integrate is to approximate the values in the range of interest with a polynomial, then integrate the polynomial
• The trapezoidal method uses first-order polynomial (straight lines) to connect adjacent points
Trapezoidal Rule
• For a function that can be evaluated throughout the range from x0 to xn, where n is the number of equal base trapezoids, we can calculate the area:
Trapezoidal Rule
• The error of approximating a curve with a straight line can be large
• To reduce error, we can increase the number of intervals, however, round off error will eventually dominate, so a reasonable number for n is 100-200
• Example 21.2
• Questions?
Simpson’s Rules
• For greater accuracy, each interval could be estimated using a higher order polynomial instead of a straight line

Simpson’s 3/8

Simpson’s 1/3

Simpson’s 1/3 Rule
• Approximating the curve with a second order Lagrange polynomial
Simpson’s 1/3 Rule
• So, this will break n points into n/2 even base parabolic intervals (n must be even)
• Example 21.5
• Questions?
Simpson’s 3/8 Rule
• If a third order Lagrange interpolating polynomial is used, 4 data points are used for each interval
• Because the accuracy is not much higher than the 1/3 rule and more points calculations are required, typically 3/8 is not used for the entire interval
Multiple ‘Rules’
• For a set of data with odd intervals, one set of four points is used with the 3/8 Rule and the remaining even number of intervals use 1/3 Rule
• This method becomes more important with unevenly spaced intervals
Newton-Cotes Formulas
• The formulas are summarized on page 604 along with the truncation error estimates
• Example 21.7, 21.8
Integration of Equations
• Romberg Integration is based on the trapezoidal rule, but it uses a recursive approach to refine the integral estimate quickly based on the values calculated by successive n=2j trapezoidal rule estimates
Integration of Equations
• This method uses a call to a trapaziod subroutine to generate estimates
• The estimates are then combined to form improved estimates with lower error n-1 times
Integration of Equations
• This process is repeated until the approximation error between successive iterations falls below a stopping criterion
• Example 22.10
• Questions?
Preparation for 28Mar