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A predictor-corrector method and stability

  1. Apr 14, 2014 #1

    wel

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    Gold Member

    A predictor-corrector method for the approximate solution of [itex]y'=f(t,y)[/itex] uses
    \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P
    \end{equation}
    as predictor and
    \begin{equation} y_{n+1}-y_{n}=\frac{h}{2}(f_{n+1}-f_{n}) \tag C
    \end{equation}
    IF [itex](P)[/itex] and [itex](C)[/itex] are used in PECE mode on the vector problem
    \begin{equation} \frac{du}{dt}=u
    \end{equation}
    \begin{equation} \frac{dv}{dt}=-10u-11v+cos(2\pi t)
    \end{equation}
    with [itex]u(0),v(0)[/itex] given, find the largest constant [itex]\gamma >0[/itex] for which the scheme is stable in the sense of Von Neumann (Fourier series stability and frequency) whenever [itex]0<\gamma<0[/itex]. Give full details of your argument.

    =>
    I haven't try very well because its really difficult question for me.
    I was thinking
    \begin{equation} y_{n+1}=y_{n}+hf_{n} \tag P
    \end{equation}
    as predictor and
    \begin{equation} y_{n+1}=y_{n}+\frac{h}{2}(f_{n+1}-f_{n}) \tag C
    \end{equation}
    iam trying to get transition matrix but these condition
    \begin{equation} \frac{du}{dt}=u
    \end{equation}
    \begin{equation} \frac{dv}{dt}=-10u-11v+cos(2\pi t)
    \end{equation}
    i don't know how and where to use.
    please help me.
     
  2. jcsd
  3. Apr 15, 2014 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Are you saying you do not know how to solve for u and v in those last two equations?
    [tex]\frac{du}{dt}= u[/tex]
    should be easy. And once you have that, the second equation if
    [tex]\frac{dv}{dt}+ 11v= -10u+ cos(2\pi t)[/tex]
    is a relatively easy "linear equation with constant coefficients".
     
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