A scale-dependent hybrid algorithm for multi-dimensional time fractional differential equations

Abstract

Time fractional differential model is an effective tool to characterize anomalous diffusion phenomena in hydrology and environmental science. Efficient numerical method is necessary to overcome the bottleneck of expensive computational cost for real-world application. Therefore, this paper proposes a scale-dependent hybrid algorithm to numerically solve multi-dimensional time fractional differential models. We employ the Hausdorff metric-based hybrid algorithm to discretize time terms of time fractional differential equations (FDEs) with variable step size, which is applicable to non-uniform time steps geological problems. Meantime, another advantage of the proposed algorithm is optimizing the computational process. The method only requires O( $$n_s$$ $$n_e$$ ) memory and O( $$n_s$$ $$n_t$$ $$n_e$$ ) computational cost while classical finite difference method relatively demands O( $$n_s$$ $$n_t$$ ) and O( $$n_s$$ $$n_t^2$$ ), where $$n_s$$ , $$n_t$$ and $$n_e$$ are the number of space nodes, time steps and exponentials, respectively. Furthermore, we adopt a meshless generalized finite difference method to discretize space terms of FDEs. Robustness and accuracy of the new algorithm are verified by two examples and a set of experimental data.