Advanced topic in numerical solution of differential equation

[ra_forever8]

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 : dy/dt = λy=f
which has exact solution of y= yo 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
-1 < 1+ h λ+ θ(hλ)^2 < 1
Kindly anyone help me after this , how to investigate the stability of PECE method

 

[jvicens]

and what are the possible conclusions from this analysis.

To investigate the stability of the PECE method, we can analyze the behavior of the solution y(n+1) as n increases. This can be done by looking at the amplification factor, which is the ratio of the solution at the next time step to the current time step.

In this case, the amplification factor is 1+ h λ+ θ(hλ)^2. If this value is less than 1, the solution will decrease over time and the method is stable. However, if the amplification factor is greater than 1, the solution will increase over time and the method is unstable.

From the analysis above, we can conclude that the stability of the PECE method depends on the values of h, λ, and θ. If h and λ are small, the method is more likely to be stable. However, if θ is close to 1, the method may become unstable.

Furthermore, we can also conclude that the stability of the PECE method is affected by the choice of the evaluation step (E). This step is used to evaluate the predictor and corrector steps and can impact the stability of the method.

In general, the PECE method is stable for most values of h, λ, and θ as long as they are chosen appropriately. However, it is important to carefully select these parameters to ensure stability and accurate results.