Question about what the brackets on the superscript on this question means?

Question

Good day :) I would just like to ask about what the brackets on the superscript on this question means? Thank you for your help.

Answer 1

The exponents seem to be the discrete stages of predictors at the k-th stage. This is the usual notation for Runge-Kutta-type predictor-corrector methods of solving differential equations. In the wikipedia article https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods these are lower indices, but if one treats systems of differential equations, the lower indices are usually reserved for the components of the vector, so the stages need to be indicated somewhere else. This should be evident from the rest of the text.

Answer 2

Another interpretation: You have some state vector $z$ containing position components $y$ and velocity components $y'$.

With some piecewise polynomial interpolation functions inserted, some fixed numerical method produces solution approximations $z(x,h)$ for step size $h$. You are to compare the approximations $z^{[k]}(x)=z(x,h/2^k)$ for some fixed reference step size.

It is also possible that you only are to compare the easier obtainable values at the points $x=x_0+jh$, $j=1,2,3,...$, where for $k>0$ the big time step is split into $2^k$ substeps.