Is this 'predictor corrector' method implicit or explicit?

Question

I was of the understanding that the 'predictor corrector' method consisted of one explicit step method and one implicit method.

I've been asked to implement the following function:

ybar_n+1 = y_n + (h/12)(23 f_n - 16 f_n-1 + 5f_n-1)

y_n+1 = y_n + (h/24)(9 fbar_n+1 + 19f_n - 5f_n-1 + f_n-1)

where f = dy/dx = -(y+1)(y+3) and fbar = f(x, ybar)

So surely I find ybar(n+1) with an explicit formula, then substitute this into fbar(n+1) and the second equation is also explicit? So is this really a predictor corrector method? Yet it supposedly combines the 3-step Adam Bashforth and Adam Moulton methods.

I've also been asked to find the order of accuracy of this new function, but it appears very similar to that of my (explicit) Adam Bashforth method. Is this a mistake on my part and it should actually be higher order?

Answer 1

Your method is based on the implicit Adams-Moulton method. However, just using the fixed PECE (E=evaluate) schema makes this indeed into an explicit method, as you observed.

One could call this implicit if the corrector step is iterated until an estimate for the error towards the converged value is (widely) below the estimate of the method step truncation error.