  In recent years, an intensive study of strong approximation of stochastic
differential equations (SDEs) with a drift coefficient that may have
discontinuities in space has begun. In many of these results it is assumed that
the drift coefficient satisfies piecewise regularity conditions and the
diffusion coefficient is Lipschitz continuous and non-degenerate at the
discontinuity points of the drift coefficient. For scalar SDEs of that type the
best $L_p$-error rate known so far for approximation of the solution at the
final time point is $3/4$ in terms of the number of evaluations of the driving
Brownian motion and it is achieved by the transformed equidistant
quasi-Milstein scheme, see [M\"uller-Gronbach, T., and Yaroslavtseva, L., A
strong order 3/4 method for SDEs with discontinuous drift coefficient, to
appear in IMA Journal of Numerical Analysis]. Recently in [M\"uller-Gronbach,
T., and Yaroslavtseva, L., Sharp lower error bounds for strong approximation of
SDEs with discontinuous drift coefficient by coupling of noise,
arXiv:2010.00915 (2020)] it has been shown that for such SDEs the $L_p$-error
rate $3/4$ can not be improved in general by no numerical method based on
evaluations of the driving Brownian motion at fixed time points. In the present
article we construct for the first time in the literature a method based on
sequential evaluations of the driving Brownian motion, which achieves an
$L_p$-error rate of at least $1$ in terms of the average number of evaluations
of the driving Brownian motion for such SDEs.
