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Consistency of Crank Nicolson method

  1. Mar 1, 2009 #1
    1. The problem statement, all variables and given/known data

    Analyze the consistency of the Crank Nicolson method:

    u_n+1 = u_n + h/2[f_n + f_n+1]

    where f_n = f(t_n, u_n)

    2. Relevant equations

    How does this method work if you have to have u_n+1 to calculate f_n+1, and vice versa? Which comes first?

    3. The attempt at a solution

    I have worked other consistency analysis problems with simpler approximations, such as Forward Euler Method, but I'm still struggling with this question in general (order, stability, consistency). I know (I think) that we want to see that the difference (error) between the approximation of the method and the actual solution goes to zero as h(the fixed interval size) goes to zero, which says the method is consistent. I have been doing that by deriving the value of f from a Taylor series expansion of u around x_n. So,

    y_n+1 = y_n + hf_n + (h^2/2)*f'_n + (h^3/6)*f''_n + . . .

    and by assuming the method is true for actual y:

    (y_n+1 - y_n)/h = (1/2)*[f_n+1 + f_n]

    Not sure where to go from there. Any help appreciated.
  2. jcsd
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