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Numerical DE: Theta method

  1. May 8, 2005 #1

    I am given the method: y_(n+1) = y_n + h f(t_n + w h, (1-w)y_n + w y_(n+1).

    I am to determine the region of absolute stability;

    I am also to determine for which w in [0, 1] is the method A(a) stable,
    i.e., the region of absolute stability contains a sector about the negative
    real axis.

    I found the root of the characteristic polynomial in the complex plane to

    z = [1 + h k (1-w) ] / [1 - h k w]

    So RAS = {hk : |z| < 1}

    Can I simply that further? I.e., can I get a more explicity formula for
    what the hk that satisfy z < 1 is?

    Also I am at a loss on how to solve for when the method is A(a) stable.
    Please help if you can. Thanks in advance.
  2. jcsd
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