i saw in stream 43 that casey used a regular integration method, something like
p'=1/2*a*t^2+vt+p
in the past, i saw an article which said that using that integration method is not great, and you should use something like runge-kute or vervet method, or something simple like averaging the velocities between start and end. is casie saving it for future episodes or he just doesn't think it's significant?
Those are numerical integration techniques, not analytical integration techniques. You always use analytic functions if you have them, because they're perfect. You only use Runge-Kutta or Verlet schemes when you have to do numerical integration, which is not perfect.
Since we're not trying to do physical simulation, we may never have to do numerical integration, and may always be able to do perfect integration, which is always strictly superior because it has literally no error besides bit rounding.
Methods like Runge-Kutta or Verlet are for solving differential equations. In the case of the motion equations, they aren't necessary, unless the acceleration depends not only on the time, but also on the current position (force field) and / or velocity (drag).
Even then, you could get away with using simple integration if the force field's strength or direction doesn't vary too much over the length dt * v(t), where dt is the current time step. In this case, you can just assume a to be constant and integration becomes trivial. Similarly for drag.
What about collision detection? That's a force field which vary significantly in vdt, anyway, i think that euler is bad even for forces that doesn't vary significantly.
Yes, force-based collision resolution is one of the cases where it can be necessary to resort to numerical integration. But, especially in 2D games with simple collision hulls, collision resolution can often be done exactly with impulse reflection instead of applying forces.
Collision reflection is like an infinite force, thats why i think you should use those methods. I think that even a 2D game, has to use an elaborate way to handle collisions, to avoid framerate dependence and speed dependence and what's not.