Answer to Quiz 11 Fri 18 Nov 2005

1. The formula

                    h  (                            )
   y(n+1) = y(n) + --- (- f(n-1) + 8 f(n) + 5 f(n+1))
                    12 (                            )
   

   is used to solve dy/dx = f(x,y(x)).
   Use the method of undetermined coefficients to derive this formula.

   Answer:

      Without loss of generality, we consider x(n) = 0,
      then x(n-1) = -h and x(n+1) = +h.
   
       h
      /
      | f(x,y(x)) dx = c(-1) f(-h) + c(0) f(0) + c(+1) f(+h)
      /
       0
   
   With three unknown coefficients, we can require that the first
   three basis functions 1, x, and x^2 are integrated exactly:
   
          h
         /
   f=1   | 1 dx =  h  = c(-1)      + c(0)   + c(+1) 
         /
         0
   
         h          2
         /         h
   f=x   | x dx = --- = c(-1) (-h) + c(0) 0 + c(+1) (+h)
         /         2
         0
   
          h         3
      2  /  2      h              2                     2
   f=x   | x dx = --- = c(-1) (-h) + c(0) 0 + c(+1) (+h)
         /         3
         0
   
   The last two equations are  -c(-1) + c(1) = h/2
                                c(-1) + c(1) = h/3
   
   Adding these two equations leads to c(1) = 5*h/12, and
   with further elimination we find c(-1) = -h/12, c(0) = 8*h/12.
   
   

2. Suppose we want to solve an initial-value problem y' = f(x,y), y(0) = y_0, with a predictor-corrector method using four points in each step. Calculate how many times we evaluate f to approximate y(1.4), using step size h = 0.2.
   Justify your count.

   Answer:

   With a fixed step size of h = 0.2, we compute y(7) at x = 1.4:
   
           4  +  4  +  4  +  1  +  1  +  1  +  1   = 16 function evaluations
    y(0)  y(1)  y(2)  y(3)  y(4)  y(5)  y(6)  y(7)
   --+-----+-----+-----+-----+-----+-----+-----+------------
    0.0   0.2   0.4   0.6   0.8   1.0   1.2   1.4
          RK    RK    RK    PC    PC    PC    PC
   
   The predictor needs 4 points, which are computed by a Runge-Kutta method.
   Since the order of the Runge-Kutta method must match the order of the
   predictor-corrector method, it must use the same number of function
   evaluations, i.e.: 4 in each step.
   
   Note that function evaluations are stored.  With a fixed step size,
   a predictor-corrector method needs only one new function evaluation
   in each step.