How Does the PECE Method Improve Numerical Solution Accuracy?

[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}
as corrector. Determine the orders of the PECE,PECECE methods, with explanation.
=>
I know the PECE ($P=P* +1$)
Order of Predictor $P=1$
Order of Corrector $C=2$
So, Order of PECE is $2$
Similarly,
PECECE ($P=P* +2$)
Order of Predictor $P=1$
Order of Corrector $C=2$
So, Order of PECECE is also $2$ (limited by order of truncation error)
CAN SOMEONE PLEASE GIVE ME BETTER EXPLANATION THAN MINE.

 

[jvicens]

Sure, let me try to provide a more detailed explanation for the orders of the PECE and PECECE methods.

First, it's important to understand what we mean by the "order" of a numerical method. The order of a method refers to the rate at which the error in the numerical solution decreases as we decrease the step size $h$. In other words, a higher order method will give us a more accurate solution for a given step size compared to a lower order method.

Now, let's look at the PECE method. The predictor step is given by equation (P) and the corrector step is given by equation (C). In the predictor step, we use the current value of $f(t,y)$ at time $t_n$ to estimate the value of $y$ at the next time step $t_{n+1}$. This is a first-order approximation, since we are only using the information at the current time step.

In the corrector step, we use a weighted average of the current and predicted values of $f(t,y)$ to improve our estimate of $y$ at $t_{n+1}$. This is a second-order approximation, since we are using information from both the current and predicted time steps.

Therefore, the overall order of the PECE method is limited by the order of the corrector step, which is 2. This means that the error in the numerical solution will decrease at a rate of $O(h^2)$ as we decrease the step size $h$.

Similarly, for the PECECE method, the predictor step is still a first-order approximation, and the corrector step is now a fourth-order approximation (since we are using information from both the current and predicted time steps, as well as the next predicted time step). However, the overall order of the method is still limited by the order of the corrector step, which is 4. This means that the error in the numerical solution will decrease at a rate of $O(h^4)$ as we decrease the step size $h$.

In conclusion, the PECE method has an overall order of 2, while the PECECE method has an overall order of 4. This means that for the same step size, the PECECE method will give us a more accurate solution compared to the PECE method.