In this paper a class of methods, called improved split-step one-leg theta methods(ISSOLTM), are introduced and are shown to be convergent for SDEs with one-sided Lipschitz continuous drift coefficient if the method parameter satisfies 1/2 ≤ θ ≤ 1. At the same time, we improve the strong order from one half to one on the basis of the existing literature. For 0 ≤ θ ≤ 1/2, under the additional linear growth condition for the drift coefficient, the methods are also strongly convergent with the the order 1. Finally, the obtained results are supported by numerical experiments.
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