In　this　paper,　numerical　methods　for　the　nonlinear　stochastic　differential　equations　(SDEs)　with　non-global　Lipschitz　drift　coefficient　are　discussed.　The　existing　known　results　have　only　so　far　shown　that　the　classical　(explicit)　Euler-Maruyama　(EM)　approximate　solutions　converge　to　the　true　solution　in　probability　[22,23].　More　recently,　the　authors　in　[16]　proved　that　the　classical　EM　method　will　diverge　in　L2　sense　for　the　underlying　SDEs　in　this　paper　(and　those　SDEs　with　superlinearly　growing　coefficients).　These　strongly　indicate　that　the　classical　EM　method　is　not　good　enough　for　the　highly　nonlinear　SDEs.　However,　in　this　paper,　we　introduce　a　modified　EM　method　using　stopping　time　and　show　successfully　that　the　discrete　version　of　the　modified　EM　approximate　solution　converges　to　the　true　solution　in　the　strong　sense　(namely　in　L2)　with　a　order　arbitrarily　close　to　a　half.　©　2013　Elsevier　Ltd.　All　rights　reserved.