Time-Splitting Schemes for Fractional Differential Equations I: Smooth Solutions

We propose three time-splitting schemes for nonlinear time-fractional differential equations with smooth solutions, where the order of the fractional derivative is 0<α<1. While one of the schemes is of order α, the other two schemes are of order 1+α and 2-α and thus they can be combined to provide flexible numerical methods with convergence order no less than $3/2$. We prove the convergence and stability of the proposed schemes. Numerical examples illustrate the flexibility and the efficiency of these time-splitting schemes and show that they work for multirate and stiff time-fractional differential systems effectively.