  A numerical search approach is used to design high-order diagonally implicit
Runge-Kutta (DIRK) schemes equipped with embedded error estimators, some of
which have identical diagonal elements (SDIRK) and explicit first stage
(ESDIRK). In each of these classes, we present new A-stable schemes of order
six (the highest order of previously known A-stable DIRK-type schemes) up to
order eight. For each order, we include one scheme that is only A-stable as
well as schemes that are L-stable, stiffly accurate, and/or have stage order
two. The latter types require more stages, but give better convergence rates
for differential-algebraic equations (DAEs), and those which have stage order
two give better accuracy for moderately stiff problems. The development of the
eighth-order schemes requires, in addition to imposing A-stability, finding
highly accurate numerical solutions for a system of 200 equations in over 100
variables, which is accomplished via a combination of global and local
optimization strategies. The accuracy, stability, and adaptive stepsize control
of the schemes are demonstrated on diverse problems.
