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Numerical instability may arise either from the stiffness of the linear operator or from the perturbation of the nonlinear drift under discretization, or both. Typical applications arise from the space discretization of an SPDE, stochastic volatility models in finance, or certain ecological models. Under conditions that include montonicity, we prove that a timestepping strategy which adapts the stepsize based on the drift alone is sufficient to control growth and to obtain strong convergence with polynomial order. The order of strong convergence of our scheme is (1 \u2212 <jats:italic>\u03b5<\/jats:italic>)\/2, for <jats:italic>\u03b5<\/jats:italic> \u2208 (0,1), where <jats:italic>\u03b5<\/jats:italic> becomes arbitrarily small as the number of finite moments available for solutions of the SDE increases. Numerically, we compare the adaptive semi-implicit method to a fully drift-implicit method and to three other explicit methods. Our numerical results show that overall the adaptive semi-implicit method is robust, efficient, and well suited as a general purpose solver.<\/jats:p>","DOI":"10.1007\/s11075-021-01131-8","type":"journal-article","created":{"date-parts":[[2021,6,1]],"date-time":"2021-06-01T04:04:11Z","timestamp":1622520251000},"page":"721-747","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Adaptive Euler methods for stochastic systems with non-globally Lipschitz coefficients"],"prefix":"10.1007","volume":"89","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9707-2822","authenticated-orcid":false,"given":"C\u00f3nall","family":"Kelly","sequence":"first","affiliation":[]},{"given":"Gabriel J.","family":"Lord","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,31]]},"reference":[{"key":"1131_CR1","first-page":"401","volume":"17","author":"J Appleby","year":"2010","unstructured":"Appleby, J., Kelly, C., Mao, X., Rodkina, A.: On the local dynamics of polynomial difference equations with fading stochastic perturbations. 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