I am aware than Euler explicit is conditionally stable, and Euler implicit is unconditionally stable. And I am aware that it is probably pointless to use Euler implicit with a small computational step $\Delta t$ for which Euler explicit is stable. But from a theoretical point of view, should one of the two be more accurate than the other, of course for $\Delta t$ for which the explicit method is stable? It can be shown that the local truncation error for both Euler explicit and Euler Implicit scales linearly with $\Delta t^2$, and therefore the global truncation error should scale with $\Delta t$, so they are both first order accurate. And from what I understand also the constant in front of $\Delta t$ should be the same, or is here where I am doing wrong? In FLOW-3D technical manual on implicit vs explicit methods I read that:
For an implicit method to have minimal under-relaxation (i.e., little damping), a time-step size much smaller than the stable, explicit value would have to be used. [...] To reduce this under-relaxation damping the time-step size would have to be much smaller than the explicit stability limit, but this makes little sense since an implicit method is not required.
Is this true? Since they are both first order, are we saying that it is possible to prove that the proportionality constant in front of $\Delta t$ is larger for the implicit scheme and depends on the step (larger damping for higher steps)? Can the answer to this question be generalized to Runge-Kutta schemes?