There are so many Runge Kutta methods, including
- Dormand-Prince 45
- Cash-Karp 54
- Fehlberge (sic) 78
Is there any comparison between them?
Well, sure. Here are some traits to compare:
- Is the method implicit or explicit? (All of your examples are explicit RK methods.)
- What is the order of convergence? Are there any embedded error estimators? How many, and of what order(s)?
- How many stages does it have?
- How many function evaluations are required? For implicit methods, how many linear solves are involved?
- What is the stability region? Is it A-stable? L-stable?
My relatively simplistic understanding of these methods is that:
- Dormand-Prince 4(5) is a 4th-order method with a 5th-order error estimator designed to minimize the error in the 5th-order solution, which is preferable when using the 5th-order solution to continue the integration
- Fehlberg methods are supposed to minimize the error in the lower-order solution (in your case, the 7th-order solution); in general, Dormand-Prince methods of the same order should perform better
- I've heard of the Cash-Karp discretization, but I haven't across a source that claims any sort of advantage or disadvantage compared with the Dormand-Prince methods
What is each approach sacrificing?
What is the general trade-off in different RK methods?
You should look at (or construct!) work-precision diagrams for the problems and integrators you're interested in investigating. Hairer and Wanner have some examples.
Which approach is suitable for what model?
Again, you should look at work-precision diagrams. To a very coarse approximation, you want explicit Runge-Kutta methods for non-stiff problems (assuming, that is, you want to use a Runge-Kutta method), and for stiff problems, you're likely to want an implicit Runge-Kutta method. Methods with a higher order of accuracy per time step are likely to perform better at stricter error tolerances than methods with a lower order of accuracy. However, the best approach is (very) problem-dependent.