In this paper, we derive an Euler-Maruyama (EM) method for a class of multi-term fractional stochastic nonlinear differential equations, and prove its strong convergence. The strong convergence order of this EM method is $\min\{\alpha_{m}-0.5,~\alpha_{m}-\alpha_{m-1}\}$, where $\{\alpha_{i}\}_{i=1}^{m}$ is the order of Caputo fractional derivative satisfying that $1>\alpha_{m}>\alpha_{m-1}>\cdots>\alpha_{2}>\alpha_{1}>0$, $\alpha_{m}>0.5$, and $\alpha_{m}+\alpha_{m-1}>1$. Then, a fast implementation of this proposed EM method is also presented based on the sum-of-exponentials approximation technique. Finally, some numerical experiments are given to verify the theoretical results and computational efficiency of our EM method.

In this paper, we present a finite difference and a compact finite difference schemes for the time fractional nonlinear diffusion-wave equations (TFNDWEs) with the space fourth order derivative. To reduce the smoothness requirement in time, the considered TFNDWEs are equivalently transformed into their partial integro-differential forms with the classical first order integrals and the Caputo derivative. The finite difference scheme is constructed by using Crank-Nicolson method combined with the midpoint formula, the weighted and shifted Gr$\ddot{u}$nwald difference formula and the second order convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and fourth order Stephenson scheme are used in spacial direction. Then, the compact finite difference scheme is developed by using the fourth order compact difference formula for the spatial direction. The stability and convergence of the proposed schemes are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to support our theoretical results.