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In this paper, we use the truncated EM method to study the finite time strongconvergence for the SDEs with Poisson jumps under the Khasminskii-typecondition. We establish the finite time $ \mathcal L ^r (r \ge 2) $ convergencerate when the drift and diffusion coefficients satisfy super-linear conditionand the jump coefficient satisfies the linear growth condition. The resultshows that the optimal $\mathcal L ^r$-convergence rate is close to $ 1/ (1 +\gamma)$, where $\gamma$ is the super-linear growth constant. This issignificantly different from the result on SDEs without jumps. When all thethree coefficients of SDEs are allowing to grow super-linearly, the $ \mathcalL^r (0<r<2)$ strong convergence results are also investigated and the optimalstrong convergence rate is shown to be not greater than $1/4$. Moreover, weprove that the truncated EM method preserve nicely the mean squareexponentially stability and asymptotic boundedness of the underlying SDEs withPiosson jumps. Several examples are given to illustrate our results.

The truncated EM method for stochastic differential equations with Poisson jumps