A Second Order Energy Dissipative Scheme for Time Fractional L$$^{2}$$ Gradient Flows using SAV Approach

In this paper, we propose and analyze a second order efficient numerical scheme for the time fractional $$L^2$$ gradient flows. The proposed scheme is based on scalar auxiliary variable (SAV) approach and L2- $$1_{\sigma }$$ time stepping with the so-called sum-of-exponentials (SOE) technique. The main idea is to split the time fractional derivative into two parts: the local part and the history part. Then, we use the L2- $$1_{\sigma }$$ formula to discretize the local part and adopt the extended SAV approach to deal with the history and nonlinear terms. The unconditional stability of the numerical scheme is rigorously proved for the uniform mesh. The main novelty of the paper is: this is the first proof of the unconditional stability of the L2- $$1_{\sigma }$$ time stepping schemes with SOE technique for time fractional $$L^{2}$$ gradient flows. Several numerical examples are provided to verify the accuracy and efficiency of the proposed scheme. Finally, the new scheme is applied to investigate the coarsening dynamics governed by the time fractional L $$^{2}$$ gradient flows.