By "relativity" you mean what? If relativistic mechanics of point particles in a given background, then it is computationally as complex as classical mechanics. If the background (i.e. gravitational and electromagnetic fields) is to be determined dynamically, then it is harder because you have infinitely many degrees of freedom. But that's the case of many non-relativistic classical systems (fluid dynamics...) too.

it seems to me that you would have to set up a system where the outcomes of relativity are explicitly stated, while the classical outcomes are implicit

What does this mean? I can think of several interpretations, all of them false:

1. It is impossible to calculate the evolution of a relativistic system from the initial state only, you also need to know the final state too.
2. It is possible to do calculate the relativistic evolution from the initial state, but you have to do it symbolically, numerical methods won't work.
3. It is impossible to calculate anything numerically in a relativistic system, solutions have to be guessed and later verified, they can't be systematically constructed.