Semi-implicit split-step numerical methods for a class of nonlinear stochastic differential equations with non-Lipschitz drift terms

In this paper, we discuss numerical solutions of a class of nonlinear stochastic differential equations using semi-implicit split-step methods. Under some monotonicity conditions on the drift term, we study moment estimates and strong convergence properties of the numerical solutions, with a focus on stochastic Ginzburg–Landau equations. Moreover, we compare the performance of various numerical methods, including the tamed Euler, truncated Euler, implicit Euler and split-step procedures. In particular, we discuss the empirical rate of convergence and the computational cost of these methods for certain parameter values of the models used.