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Math Help - Euler's implisit - convergence of each step

  1. #1
    Member Mollier's Avatar
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    Euler's implisit - convergence of each step

    Hi,

    problem:

    the trapezoid method of solving differential equations is,

    y(t_{i+1}) = y(t_i) + h\frac{f(y(t_{i+1})) + f(y(t_i))}{2}.

    Since the method is implicit, we must solve for y(t_{i+1}) on each step by some iterative method. If we use the fixed point iteration x=g(x), we know that the method will converge if |g'(x)|<1 for x\in[y(t_i),y(t_{i+1})].

    Give a condition on h such that the fixed point iteration converges at each step.

    attempt:
    First of all, I need to write down an expression for g(x). Fixed-point iteration needs an initial guess. I use y(t_i) as my initial guess and write g as,

    g(x) = x + h\frac{f(x)+f(x)}{2} = x + hf(x).

    But now

    g'(x) = 1 + hf'(x) < 1 \Rightarrow h<0,

    which doesn't make much sense.
    What am I doing wrong?

    Thanks
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  2. #2
    Member Mollier's Avatar
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    I think that notation might have me fooled.
    At each step, y(t_i) and f(y(t_i)) are constants and so,

    |g'(t)| = |\frac{h}{2}f'(y(t))| < |1|.

    This means that,

    h < \frac{2}{f'(y(t))},

    at each step in order to be sure that fixed-point iteration will converge.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Mollier View Post
    Hi,

    problem:

    the trapezoid method of solving differential equations is,

    y(t_{i+1}) = y(t_i) + h\frac{f(y(t_{i+1})) + f(y(t_i))}{2}.

    Since the method is implicit, we must solve for y(t_{i+1}) on each step by some iterative method. If we use the fixed point iteration x=g(x), we know that the method will converge if |g'(x)|<1 for x\in[y(t_i),y(t_{i+1})].

    Give a condition on h such that the fixed point iteration converges at each step.

    attempt:
    First of all, I need to write down an expression for g(x). Fixed-point iteration needs an initial guess. I use y(t_i) as my initial guess and write g as,

    g(x) = x + h\frac{f(x)+f(x)}{2} = x + hf(x).

    But now

    g'(x) = 1 + hf'(x) < 1 \Rightarrow h<0,

    which doesn't make much sense.
    What am I doing wrong?

    Thanks
    Try the itteration on $$u_k:

    u_{k+1}=y(x_i)+h \dfrac{f(u_k)+f(x_i)}{2}

    and start with u_0=y(x_i)+hf(x_i) (which is a standard explicit Euler step)

    This appears to be equivalent to the 1st order itterated predictor-corrector method.

    CB
    Last edited by CaptainBlack; December 13th 2010 at 11:21 PM.
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