Results 1 to 3 of 3

Math Help - Implicit Euler Scheme and stability

  1. #1
    Junior Member
    Joined
    Mar 2014
    From
    uk
    Posts
    53

    Implicit Euler Scheme and stability

    Find the fixed points of the implicit Euler scheme
    \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1})
    \end{equation}
    when applied to the differential equation $y'=y(1-y)$ and investigate their stability?
    =>
    implicit Euler scheme
    \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1})
    \end{equation}
    $y'=y(1-y)$
    \begin{equation} y_{n+1}=y_{n}+hy_{n+1}(1-y_{n+1})
    \end{equation}
    \begin{equation} y_{n+1}=y_{n}+hy_{n+1}-hy^2_{n+1}
    \end{equation}
    For fixed points
    $y_{n+1}=y_{n}$
    \begin{equation} y_{n}=y_{n}+hy_{n}-hy^2_{n}
    \end{equation}
    $y_{n}=0$ or $1$
    I got problem with stability but this is what I have done
    $y_{n}= \alpha +\epsilon^n$, $y_{n+1}= \alpha +\epsilon^{n+1}$,
    \begin{equation} \alpha +\epsilon^{n+1}= \alpha +\epsilon^n + h (\alpha +\epsilon^{n+1})(1-\alpha -\epsilon^{n+1}) \end{equation}
    \begin{equation} \epsilon^{n+1}= \epsilon^n + h (\alpha +\epsilon^{n+1})(1-\alpha -\epsilon^{n+1}) \end{equation}
    When $y_{n}=0=\alpha$
    \begin{equation} \epsilon^{n+1}= \epsilon^n + h \epsilon^{n+1}(1-\epsilon^{n+1}) \end{equation}
    I don't what to say or do after that to determine the stability.
    When $y_{n}=1=\alpha$
    \begin{equation} \epsilon^{n+1}= \epsilon^n - h \epsilon^{n+1}(1+\epsilon^{n+1}) \end{equation}
    same again what can say about with my answer to investigate the stability.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,652
    Thanks
    1063

    Re: Implicit Euler Scheme and stability

    what makes a stationary point stable vs. unstable? (hint think 2nd derivative)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2014
    From
    uk
    Posts
    53

    Re: Implicit Euler Scheme and stability

    h>0 unstable
    h<0 stable
    but how to get into that form. that's where I got stuck.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: November 29th 2011, 03:59 PM
  2. Proving Stability and Asymptotic Stability of Homogeneous Equations
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: November 4th 2010, 12:16 PM
  3. Stability of forward Euler
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: September 15th 2010, 10:00 PM
  4. Replies: 4
    Last Post: October 31st 2009, 01:22 AM
  5. Implicit Euler
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 18th 2009, 02:42 AM

Search Tags


/mathhelpforum @mathhelpforum