Hi all,

I have the following question regarding the use of Embedded Runge-Kutta (RK) method for the sake of integrating a differential equation with a linear friction term:
$dy/dt = f(t,y) - v y.$

A useful trick is to consider instead the system
$dx/dt = g(t,x)$
where:
$x = \exp(v t) y$
and
$g(t,x) = \exp(v t)f(t, \exp(-v t) x).$

One then applies the usual RK schemes to $x$ and $g$ and tranposes the solution back to $y$ and $f$.

In order for this substitution to be applicable to RK schemes, it is necessary that the sequence of time increments $\{a_i\}_i$ be increasing. This is the case for the 4th order formula where the coefficients of the time increments are respectively 0, 1/2, 1/2 and 1. Unfortunately, this condition is violated by the Cash-Karp parameters for the 5th order embedded RK method ($a_5 = 1 > a_6 = 7/8)! The original Fehlberg parameters also fail to satisfy this condition.

Is there another set of parameters where this condition is satisfied?

Thanks for your time,
Tomo