Investigate the stability of the PECE method where
P=Predictor : $y_{n+1} = y_n + hf(y_n)$
C=Corrector: $y_{n+1} = y_n + h [(1-θ) f(y_n) + θ f(y_(n+1))], (0<θ<1) $
and E is the evaluation step.
=> substituting the predictor into corrector gives: $$ y_{n+1} = y_n + h [(1-θ) f(y_n) + θf( y_n+ h f y_n )]\ \ \ \ (1) $$ stability : $\frac{d}{dt}y = λy=f$
which has exact solution of $y(t)= y_0 e^{λt}$.
Substitute $f=λy$ into $(1)$: $$ y_{n+1} = y_n + h [(1-θ) λ y_n + θλ( y_n+ h λy_n )] $$ $$ y_{n+1} = y_n + h ( λ y_n-θ λ y_n + θλ y_n+ θh λ^2 y_n ) $$ $$ y_{n+1} = y_n + h ( λ y_n+ θh λ^2 y_n ) $$ $$ y_{n+1} = y_n + h λ y_n+ θ(h λ)^2 y_n $$ $$ y_{n+1} = y_n (1+ h λ+ θ(hλ)^2) $$ now,
$$|1+ h λ+ θ(hλ)^2|<1$$ So: $$ -1 < 1+ h λ+ θ(hλ)^2 < 1 $$
Kindly anyone help me after this , how to investigate the stability of PECE method
Denote $z=h\lambda$, and solve the second order equation in $z$: $$ 1+z+\theta z^2<1, $$ $$ 1+z+\theta z^2>-1, $$ Then, sub $z=h\lambda$ and find $\theta$. You will find an interval within wich you method will be stable.