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In previous work, we proposed a methodology for the formal modeling, simulation, and model checking of interacting hybrid systems in the rewriting-logic-based Real-Time Maude tool. In that effort/flow-inspired methodology, both the physical components and their interactions are explicitly modeled, so that one only needs to describe the dynamics of single components and interactions, instead of having to perform the hard task of defining the continuous dynamics of the entire system. We previously used the Euler method to approximate the continuous dynamics defined by ordinary differential equations (ODEs). This paper explains (i) how we adapt, and then formalize in Real-Time Maude, the more precise Runge-Kutta numerical approximation methods to the effort/flow approach, in particular to coupled ODEs; and (ii) how we can use adaptive time increments to better approximate the time at which a discrete event must take place. Finally, we compare the results and the execution times for Real-Time Maude simulation and model checking with our previous approach on some thermal systems.