A Superlinearly Convergent Scheme for Multi-term Time-fractional Nonlinear Diffusion Equations with Initial Singularity

In this paper, we consider a class of multi-term time-fractional nonlinear diffusion equations (MTFNDEs) with initial boundary value conditions. Due to the initial singularity of the solutions of MTFNDEs, many existing numerical methods suffer from order reduction. To overcome this challenge, we derive a new scheme with $\min\{(\delta + \alpha_m - \alpha_{m-1})/\beta, 2\}$-order accuracy in time for $0 < \alpha_{m-1} < \alpha_m \leq 1$ and $0 < \delta < 1$ by combining the technique of variable transformation $t = s^{1/\beta}$ ($0 < \beta \leq 1$) and the linear interpolation. Meanwhile, the unconditional stability and convergence of the scheme are proved through the Fourier analysis method. Finally, numerical experiments have been given to support the theoretical results and efficiency of our proposed scheme.