IX 1

1
Numerical Integration of Differential Equations

• M. Peter Jurkat
• CS452/Mgt532 Simulation for Managerial Decisions
• The Robert O. Anderson Schools of Management
• University of New Mexico

2
Simulating Differential Equations

• Based on Eulers method yt yt-1 hyt-1
• Simple, can be grossly inaccurate
• Many useful methods most used are
• Simpsons Rule
• Runge-Kutta (RK) type methods single step,
  stable for small steps
• Predictor Corrector Methods multi-step, stable,
  need Simpsons rule or RK methods for starting
  values
• All improved by step size adjustment
• smaller for steeply rising or falling response
  surfaces
• increase step size for speed

3
Order of DEs

• Level of highest derivative in DE is order of
  system
• All numerical integration methods work with first
  order equations contain only the first
  derivative higher order equations converted to
  systems of first order

4
First Order Differential Equations

• Solve equation for first derivative yields
• y(x) dy/dx f(x, y(x) a, b, ) with y(0)
  y0
• Make x discrete by h Dx and approximate
• solve for next value of y by
• y(x h) y(x) h f(x, y(x) a, b, )
• When x is time designate the current time
  period by t then t1 t h, t-1 t h, t-2
  t 2h, get difference equation yt1 yt
  hf
• See LogisticDE.xls

5
Higher Order DE

• Use linear second order DE ay by cy
  g(t) as an example solving for y yields
• Converts into a system of two first order DEs by
  y1 y and y2 y to get y1 y2
• See EulersMethod.xls

6
Eulers Method Revisited

• Eulers method applied to y -y (oscillator)
  shows unbounded error compared to exact solution
  due to large step size, h, relative to
  oscillations step size under our control
• Assignment
• (a) Determine the largest h that will "closely"
  simulate a system modeled by y -y with initial
  values y0 0 and y0 1 over the range of 0,
  4p using Eulers method use your judgment as to
  how closely "closely" really is
• (b) What are some useful numerical measures for
  "closely" ?

7
Order ofNumerical Integration Methods

• Eulers method is a first order method it will
  integrate the differential equation of a straight
  line exactly not bad for monotonic functions
  over short distances
• Higher order methods needed for higher order DEs
• Simpsons rule is third order exact for cubics
• Runge-Kutta-Gill method is fourth order
• Most used predictor-corrector methods also fourth
  order
• Ways exist to develop any order method see most
  numerical analysis books (e.g., Hamming)

8
Runge-Kutta-Gill Method

• Given by
• Where
• Weighted average of 4 estimates of the derivative
  on the interval tn, tn1 one at beginning
  (tn), two in the middle (tn h/2), and one at end
  (tn1 tnh)
• See RKGMethod.xls applied to Normal Distribution
  density function

9
RKG (cont.)

• RK methods are an excellent compromise between
  accuracy and amount of computation
• higher than fourth order equations are rare
  enough that it does not seem worthwhile to create
  general purpose simulation programs based on
  higher than fourth order methods
• Common improvement over RK methods with fixed
  step size are methods which adjust the step size
  h depending on the rate of change of the function
• that is, h is made small in regions where f is
  large and made larger when the solution enters
  regions where f is small

10
Predictor-Corrector (PC) Methods

• Even with variable step features, RK methods can
  still take a long time - needs 4 full function
  evaluations per time step
• PC methods use 2 function evaluations -
  information needed gained by using previously
  calculated solution values
• Results in considerably faster simulation
• PC methods need RK type methods for starting and
  restarting

11
Fourth Order Milne Method Predictor

• Predictor is
• Where Li is the slope at the end of
• the current step
• one step ago
• two steps ago
• NB need three 4 prior values yn, yn-1, yn-2 ,
  and yn-3 gt fourth order method

12
Fourth OrderMilne Method Corrector

• Simpsons formula
• Where
• See RKGMethod.xls for PC applied to Normal
  Distribution density function
• Assignment Repeat CSS Assignment 1 using the RKG
  and Predictor-Corrector methods - use the same
  measure for "closely"