Runge–Kutta method with equation dependent coefficients

The family of the simplest three-stage explicit Runge–Kutta methods is examined by a conveniently adapted form of the exponential fitting approach. We obtain versions whose unusual feature is that their coefficients are no longer constant, as in the standard version, but depend on the equation to be solved. Two mathematical properties of the new versions are specially helpful for applications. Firstly, although in general the order is three, that is the same as for the standard method, this can be easily increased to four by a suitable choice of the position of the stage abscissas. Secondly, the stability properties are massively enhanced. In particular, two versions of order four are A-stable, a fact which is quite unusual for explicit methods.

► Three-stage explicit Runge–Kutta method is examined by exponential fitting approach.
► Versions with equation dependent coefficients are obtained in this way.
► These versions are much better for error and stability than the old ones.
► Thus, two of them are A-stable, a surprising property for explicit methods.