  In this work we consider a mixed precision approach to accelerate the
implemetation of multi-stage methods. We show that Runge-Kutta methods can be
designed so that certain costly intermediate computations can be performed as a
lower-precision computation without adversely impacting the accuracy of the
overall solution. In particular, a properly designed Runge-Kutta method will
damp out the errors committed in the initial stages. This is of particular
interest when we consider implicit Runge-Kutta methods. In such cases, the
implicit computation of the stage values can be considerably faster if the
solution can be of lower precision (or, equivalently, have a lower tolerance).
We provide a general theoretical additive framework for designing mixed
precision Runge-Kutta methods, and use this framework to derive order
conditions for such methods. Next, we show how using this approach allows us to
leverage low precision computation of the implicit solver while retaining high
precision in the overall method. We present the behavior of some
mixed-precision implicit Runge-Kutta methods through numerical studies, and
demonstrate how the numerical results match with the theoretical framework.
This novel mixed-precision implicit Runge-Kutta framework opens the door to the
design of many such methods.
