I've read an article (Long-term integrations and stability of planetary orbits in our Solar system) in which the authors solved the problem of the absence of an analytical solution for the solar system planetary dinamics by making a very accurate solution. They called it a "pseudo-exact solution". They then used this solution to estimate the error in planetary longitudes of their main solution, which was less precise. For instance the time step of the "exact" solution was 0.125 days, while the time step of the main solution was 8 days, meaning that the latter was 64 times bigger than the one for "exact" solution.
What I don't understand is how they could compare the two solutions since the time steps was so different. I mean, maybe they did something like
(u - y(1:64:N))/(2^p - 1)? Where
u is the main solution,
y the precise solution and
N is the number of iterations. But in that case how they could know that the terms of
y to compare with
u was precisely every 64 steps? I mean, the second element of
u for example could be compared with a term between the index 2 and 64 of
y actually. How can be chosen a specific value between 2 and 64 without checking the behaviour of the exact solution in every 64-long interval?