I’m trying to find a Runge–Kutta integrator that only requires the absolute minimum of storage, i.e. it fulfills one of the following three, presumably equivalent criteria:
Each evaluation of the derivative should only depend on the previous stage.
It stores only store three vectors, namely the beginning of the step, an intermediate temporary copy, and the end of the step, which is incrementally updated each stage (the temporary copy can then be overwritten in each stage).
It has a “diagonal Butcher tableau”.
The classical fourth-order Runge–Kutta method (RK4) is an example of this, but I am struggling to find such methods for higher orders. Do they exist?