I am currently investigating numerical integration applied to physics simulation and I have a question regarding the implicit integration of external forces.
My system is a simple particle. Depending on whether my integration scheme is explicit or implicit, my equation will involve f_n-1, the force at the previous step (for explicit integration), or f_n+1, the force at the next step (for implicit integration), hence we can write:
explicit: x_n+1 = 2x_n - x_n-1 + f_n-1 / m * dt^2
implicit: x_n+1 = 2x_n - x_n-1 + f_n+1 / m * dt^2
If the particle is tied to a linear damped spring then you can express force f with respect to x, and you can solve for x_n+1 (of course, the exact expression of force f(x, v) depends on the chosen integration scheme). This is all fine to me.
Now imagine you do not know your particle is tied to a damped spring and the only thing you have is an external force.
So my preliminary question is : in case you use an external force you are no longer able to express f_n+1 but can only know about f_n, is that right?
And my main question is: when you use an external force in an implicit integration scheme, as you no longer use f_n+1 but f_n, do you still benefit out of the full power of implicit integration? As you use neither f_n+1, nor f_n-1 but f_n, do you get something not as bad as explicit integration but not as good as implicit?