A predictor-corrector method and stability

[ra_forever8]

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ 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 $(P)$ and $(C)$ 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 $u(0)$,$v(0)$ given, find the largest constant $\gamma >0$ for which the scheme is stable in the sense of Von Neumann (Fourier series stability and frequency) whenever $0<\gamma<0$. 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.

 

[jvicens]

Thank you for your question. The predictor-corrector method is a commonly used technique for solving initial value problems in differential equations. In this method, the solution at each time step is approximated using a combination of a predicted value and a corrected value.

In the given problem, we are asked to find the largest constant $\gamma >0$ for which the scheme is stable in the sense of Von Neumann. To do this, we need to analyze the stability of the method using the Fourier series stability and frequency.

First, let us consider the predictor step given by equation $(P)$:
\begin{equation}
y_{n+1}=y_{n}+hf_{n} \tag P
\end{equation}

Substituting the given equations for $u$ and $v$ into equation $(P)$, we get:
\begin{equation}
u_{n+1}=u_{n}+hu_{n} \tag P1
\end{equation}
\begin{equation}
v_{n+1}=v_{n}+hv_{n}-10hu_{n}-11hv_{n}+hcos(2\pi t) \tag P2
\end{equation}

Now, let us consider the corrector step given by equation $(C)$:
\begin{equation}
y_{n+1}=y_{n}+\frac{h}{2}(f_{n+1}-f_{n}) \tag C
\end{equation}

Substituting the equations for $u$ and $v$ into equation $(C)$, we get:
\begin{equation}
u_{n+1}=u_{n}+\frac{h}{2}(u_{n+1}-u_{n}) \tag C1
\end{equation}
\begin{equation}
v_{n+1}=v_{n}+\frac{h}{2}(v_{n+1}-v_{n}-10(u_{n+1}-u_{n})-11(v_{n+1}-v_{n})+cos(2\pi t)) \tag C2
\end{equation}

Now, we need to find the transition matrix for the predictor step $(P)$ and the corrector step $(C)$. The transition matrix is defined as the matrix that maps the solution at time