An Explicit Two-Stage Truncated Runge–Kutta Method for Nonlinear Stochastic Differential Equations

In this paper, we construct a two-stage truncated Runge–Kutta (TSRK2) method for highly nonlinear stochastic differential equations (SDEs) with non-global Lipschitz coefficients. TSRK2 is an explicit method and includes some free parameters that can extend the accuracy of the results and stability regions. We show that this method can achieve a strong convergence rate arbitrarily close to half under local Lipschitz and Khasiminskii conditions. We study the mean square stability properties (MS-stability) of the method based on a scalar linear test equation with multiplicative noise, and the advantages of our results are highlighted by comparing them with those of the truncated Euler–Maruyama method. We also analyze the asymptotic stability properties of the method. We show that the proposed method can preserve the asymptotic stability of the original system under mild conditions. Finally, we report some numerical experiments to illustrate the effectiveness of the proposed method. As a result, we show that the new method has good properties not only in terms of practical errors but also in terms of stability.