  We present a new pathwise approximation scheme for stochastic differential
equations driven by multidimensional Brownian motion which does not require the
simulation of L\'{e}vy area and has a Wasserstein convergence rate better than
the Euler scheme's strong error rate of $O(\sqrt{h})$, where $h$ is the
step-size. By using rough path theory we avoid imposing any non-degenerate
H\"{o}rmander or ellipticity assumptions on the vector fields of the SDE, in
contrast to the similar papers of Alfonsi, Davie, and Malliavin et al. The
scheme is based on the log-ODE method with the L\'{e}vy area increments
replaced by Gaussian approximations with the same covariance structure. The
Wasserstein coupling is achieved by making small changes to the argument of
Davie, the latter being an extension of the Koml\'{o}s-Major-Tusn\'{a}dy
Theorem. We prove that the convergence of the scheme in the Wasserstein metric
is of the order $O(h^{1-2/\gamma-\varepsilon})$ when the vector fields are
$\gamma$-Lipschitz in the sense of Stein.
