- #1
wel
Gold Member
- 36
- 0
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.
\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.