  In this paper we study strong approximation of the solution of a scalar
stochastic differential equation (SDE) at the final time in the case when the
drift coefficient may have discontinuities in space. Recently it has been shown
in [M\"uller-Gronbach, T., and Yaroslavtseva, L., On the performance of the
Euler-Maruyama scheme for SDEs with discontinuous drift coefficient,
arXiv:1809.08423 (2018)] that for scalar SDEs with a piecewise Lipschitz drift
coefficient and a Lipschitz diffusion coefficient that is non-zero at the
discontinuity points of the drift coefficient the classical Euler-Maruyama
scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\in
[1,\infty)$. Up to now this was the best $L_p$-error rate available in the
literature for equations of that type. In the present paper we construct a
method based on finitely many evaluations of the driving Brownian motion that
even achieves an $L_p$-error rate of at least $3/4$ for all $p\in [1,\infty)$
under additional piecewise smoothness assumptions on the coefficients. To
obtain this result we prove in particular that a quasi-Milstein scheme achieves
an $L_p$-error rate of at least $3/4$ in the case of coefficients that are both
Lipschitz continuous and piecewise differentiable with Lipschitz continuous
derivatives, which is of interest in itself.
