  We propose new jump-adapted weak approximation schemes for stochastic
differential equations driven by pure-jump L\'evy processes. The idea is to
replace the driving L\'evy process $Z$ with a finite intensity process which
has the same L\'evy measure outside a neighborhood of zero and matches a given
number of moments of $Z$. By matching 3 moments we construct a scheme which
works for all L\'evy measures and is superior to the existing approaches both
in terms of convergence rates and easiness of implementation. In the case of
L\'evy processes with stable-like behavior of small jumps, we construct schemes
with arbitrarily high rates of convergence by matching a sufficiently large
number of moments.
