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LECTURE 11 (10/05/00)

10.2. Heun's method
	If we choose a small interval [, ] and .
	  (approximate intergal)
	Use what we learned from Euler
		
	Hence the Heun's method:
		
					^^^^^^^^^^^^^^^^^^^
	In general, to approximate Heun's method
	
	Accuracy of the method (final global error):
		
	2nd order accurate.
	e.g.
		Solve u' = (t - u)/2 with u(0) = 1
	Exact result: .
		Using h = 1, 1/2, 1/4, 1/8, we get
		(Obviously, the smaller "h", the better accuracy.)
	t_k	h=1	h=1/2	h=1/4	h=1/8	Exact
	1	.875	.831	.822	.820	.8196
	2	1.172	1.117	1.107	1.104	1.1036
	3	1.732	1.682	1.672	1.670	1.6694


10.3. Taylor series method:
	Taylor series:
	
	u' 	= f
                u''           
	u'''	= ...

	Taylor series method of order N is accurate of .
	e.g.
		Solve u' = (t - u)/2 with u(0) = 1 	(learn to compare)
	t_k	h=1	h=1/2	h=1/4	h=1/8	Exact
	1	.820	.8196	.8196	.8196	.8196
	2	1.1045	1.1037	1.1036	1.1036	1.1036
	3	1.6702	1.6694	1.6694	1.6694	1.6694

	This method is very accurate, but tedious (particularly the machine
	has to figure out the derivatives.)


10.4. Runge-Kutta method : (of order 4 is the most popular!)
	Recall the Heun's method
	
	If we define
		
	then, 
		
		
	, This is the so-called 2nd order RK----RK2.

	This method is derived by matching coefficients in Taylor series
	with a general form for ODE solver:

		
		
		
		
	After matching coefficients, We get,
			(no free parameters)
   
   

	The RK4:
       
	e.g.
		Solve u' = (t u)/2 with u(0) = 1 	(learn to compare)
	t_k	h=1	h=1/2	h=1/4	h=1/8	Exact
	1	.820	.8196	.8196	.8196	.8196
	2	1.1045	1.1037	1.1036	1.1036	1.1036
	3	1.6702	1.6694	1.6694	1.6694	1.6694
	VERY similar to the 4th order Taylor series method. Why?


10.5. Predictor-corrector (we are using all old results.)
	The orginal equation can be written as
      
	We can get approximately, 
        
	We can assume f(t,u) is a polynomial, then we can get
	the coefficients for the above equation and the solution of the IVP
		
		          
	This is something we knew already from Taylor.
	A newer formula:
		
	A problem with this formula:
		We are computing , how do we know it?
	we predict one and then correct the prediction.

Last Modified: 10/08/00