  We develop and use a novel mixed-precision weighted essentially
non-oscillatory (WENO) method for solving the Teukolsky equation, which arises
when modeling perturbations of Kerr black holes. We show that WENO methods
outperform higher-order finite-difference methods, standard in the
discretization of the Teukolsky equation, due to the need to add dissipation
for stability purposes in the latter. In particular, as the WENO scheme uses no
additional dissipation it is well-suited for scenarios requiring long-time
evolution such as the study of Price tails and gravitational wave emission from
extreme mass ratio binaries. In the mixed-precision approach, the expensive
computation of the WENO weights is performed in reduced floating-point
precision that results in a significant speedup factor of 3.3. In addition, we
use state-of-the-art Nvidia general-purpose graphics processing units and
cluster parallelism to further accelerate the WENO computations. Our optimized
WENO solver can be used to quickly generate accurate results of significance in
the field of black hole and gravitational wave physics. We apply our solver to
study the behavior of the Aretakis charge -- a conserved quantity, that if
detected by a gravitational wave observatory like LIGO/Virgo would prove the
existence of extremal black holes.
