Abstract
The fractional reaction–diffusion equation has profound physical and engineering background, and its rapid solution research is of important scientific significance and engineering application value. In this paper, we propose a parallel computing method of mixed difference scheme for time fractional reaction–diffusion equation and construct a class of improved alternating segment Crank–Nicolson (IASC–N) difference schemes. The class of parallel difference schemes constructed in this paper, based on the classical Crank–Nicolson (C–N) scheme and classical explicit and implicit schemes, combines with alternating segment techniques. We illustrate the unique existence, unconditional stability, and convergence of the parallel difference scheme solution theoretically. Numerical experiments verify the theoretical analysis, which shows that the IASC–N scheme has second order spatial accuracy and 2-alpha order temporal accuracy, and the computational efficiency is greatly improved compared with the implicit scheme and C–N scheme. The IASC–N scheme has ideal computation accuracy and obvious parallel computing properties, showing that the IASC–N parallel difference method is effective for solving time fractional reaction–diffusion equation.
Highlights
The fractional reaction–diffusion equation has a profound physical background and rich theoretical connotation
The series corresponding to these functions converge slowly, and the calculation of these special functions is quite difficult in practical applications
Even with high-performance computers, it is difficult to simulate in long-term history or large computational domain (Guo et al 2015; Sabatier et al 2014) [10, 11]
Summary
The fractional reaction–diffusion equation has a profound physical background and rich theoretical connotation. For time fractional reaction–diffusion equation, Liu et al (2015) [19] proposed an H1-Galerkin mixed finite element method and obtained the numerical results with optimal time and spatial convergence order. Chen et al (2016) [20] discussed the numerical solution of distribution order time fractional reaction–diffusion equation in semi-infinite spatial domain and proposed a fully discrete scheme based on finite difference method in time domain and spectral approximation using Laguerre function in spatial domain. Wu et al (2018) [37] proposed an alternating segment Crank–Nicolson parallel difference scheme for time fractional sub-diffusion equation, which had ideal computing accuracy and efficiency. Fu and Wang (2019) [39] developed a fast parallel finite difference method for spacetime fractional partial differential equations, which used a matrix-free preconditioned fast Krylov subspace iterative solver at each time step, significantly reducing computational complexity and memory requirement.
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