We study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition, we numerically investigate the residual diffusion phenomenon in chaotic advection. Instead of solving the Fokker-Planck equation in the Eulerian formulation, we compute the motion of particles in the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). We propose effective numerical integrators based on a splitting method to solve the corresponding SDEs. We provide rigorous error analysis for the new numerical integrators and show that our method outperforms standard Euler-based integrators. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several typical chaotic and stochastic flow (especially 3D) problems of physical interests. The existence of residual diffusivity for these flow problems is also investigated.