A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model

Abstract

Multi-term time fractional diffusion model is not only an important physical subject, but also a practical problem commonly involved in engineering. In this paper, we apply the alternating segment technique to combine the classical explicit and implicit schemes, and propose a parallel nature difference method alternating segment pure explicit-implicit (PASE-I) and alternating segment pure implicit-explicit (PASI-E) difference schemes for multi-term time fractional order diffusion equations. The existence and uniqueness of the solutions are proved, and stability and convergence analysis of the two schemes are also given. Theoretical analyses and numerical experiments show that the PASE-I and PASI-E schemes are unconditionally stable and satisfy second-order accuracy in spatial precision and 2 − α order in time precision. When the computational accuracy is equivalent, the CPU time of the two schemes are reduced by up to 2/3 compared with the classical implicit difference method. It indicates that the PASE-I and PASI-E parallel difference methods are efficient and feasible for solving multi-term time fractional diffusion equations.

Parallel Direct and Iterative Methods for Solving the Time-Fractional Diffusion Equation on Multicore Processors

“…This is caused by too much parallel overhead (time required for synchronizing the threads) for the small sizes of the SLAE. The results are more indicative for the large spatial grid m = 32 × 2 20 (roughly 32 millions) in the last experiment. Here, the parallel sweep algorithm for solving the SLAE shows better time than the classic serial Thomas algorithm.…”

“…The combination of parameters β, γ for the AOR method is β = 1.99, γ = 1.9. For the next experiments, we use the large spatial grid m = 32 × 2 20 for the same problem. The time grid in this experiment is just N = 64.…”

“…The promising way to solve various compute-intensive problems is parallel computing [12][13][14][15][16][17][18]. Several parallel algorithms has been developed specifically for the fractional differential equations and anomalous diffusion problems [19,20].…”

“…Results of experiments for Problem 2 using the logarithmic memory approach with grid m = 32 × 220 , N = 64.…”

A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation

“…It is worth pointing out that other approaches have been employed successfully to solve numerically time-fractional diffusive equations. As an example, some implicit-explicit (IMEX) difference methods have been proposed to solve time-fractional subdiffusion equations [19], reaction-diffusion equations [20], multi-term time-fractional diffusion models [21], multidimensional fractional diffusion equations [22] and nonlinear stiff fractional partial differential equations [23].…”