September 15, 2005 --- Class 6 --- Euler-Richardson Algorithm; Multiple Plots;
	A Body Falling in a Position Dependent Potential  

Activities:

	Motivation for Euler-Richardson Algorithm

		We discussed how a numerical integral done via a Taylor
		series expansion about the midpoint of the interval results
		in an integration formula with no (dt)^2 term.  This was
		used to motivate the Euler-Richardson algorithm that uses
		the acceleration and velocity at the midpoint of the time
		interval to update velocity and position through the whole
		time interval.

		You will get to implement the Euler-Richardson algorithm for
		homework.  There is a program in the textbook.

	Another use for parentheses--- multiple axis graphs on one page
		
		In ~sg/three_graphs.csh is a shell script that uses axis to
		print three graphs on a single page.
	#
	(awk '{print $1,$2}' fall.dat |axis h .3 lx t ly y ; \
	awk '{print $1,$3}' fall.dat |axis h .3 u .33 ly v -s ; \
	awk '{print $1,$4}' fall.dat |axis h .3 u .66 ly a -s ) \
  	| plot -T x -s
		
		In this case there are three awk and axis commands in one set of
		parentheses.  The awk commands pick out different columns from
		the same data file.  The axis commands make each graph shorter
		with the -h command and graphs 2 and 3 are raised 1/3 and 2/3
		of the page height with the -u option.  Note that the second
		two graphs must use the -s option to save the page.  Also you
		must add the -s option to plot, or it will complain or start
		a second plot window. (I have seen both on different systems.)
		The parentheses assure that all of the output from each 
		axis command is sent as a single stream to plot, so there is 
		only one xplot window with all three graphs.

	Discussion of an object falling in a position dependent potential

		We wish to solve is problem 3.4 of CSM.  We drop a ball
		from high above the surface of the earth (neglecting air
		resistance!).  From how great a height must we drop it so
		that it's velocity when it hits the ground is 1% different
		from what it would be assuming constant acceleration.

		There are several potential sources of error:
		1) We know from the coffee cooling problem that when we do
		numerical integration there can be a step-size error.  So
		we must be careful about this.

		2) When we integrate the equations in time, we won't
		necessarily stop integrating exactly when the object
		reaches height 0.  We may have to interpolate to find the
		velocity when the height is zero.

		3) We also must find what height results in a 1%
		difference.  Do we guess initial heights?  How do we
		accurately find the height that corresponds to 1%?

	Step size error

		We have three approaches to the step size error issue.
		One possibility is to vary the step size and see how the
		answer varies.  The accuracy on the velocity at height=0
		must be much greater than 1%.  The accuracy will depend on
		height and step size.

		We can also try to find a better method than Euler or
		Euler-Cromer.  These are each first order methods.  The
		Euler-Richardson method is one order higher.  Try running
		this code with different step sizes.  Note the accuracy of
		the energy and the position.  (Interestingly, the
		difference is negligible for the velocity.)

		Another scheme involves using the Euler-Cromer algorithm
		with different step sizes to extrapolate to zero step size.
		This was described on the board and is coded in
		adapt_step.c.  This code will increase or decrease the step
		size to achieve a desired level of accuracy during the
		integration.

		We will examine the code in adapt_step.c in my next lecture.