An efficient parallel algorithm for Caputo fractional reaction-diffusion equation with implicit finite-difference method

2020

Mathematics

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.

Section: Introductionmentioning

confidence: 99%

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

2020

Mathematics

“…Sweilam et al applied preconditioned conjugate gradient method was used to solve discrete algebraic equations in parallel, based on Crank-Nicolson difference schemes for time fractional parabolic equations [28]. Wang et al studied the parallel algorithm of implicit difference schemes for fractional reaction-diffusion equations [29]. The algorithm is based on the principle of minimizing communication, allocating computing tasks reasonably, and not changing the original serial difference schemes as much as possible.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Alternating Segment Parallel Difference Method for the Time Fractional Telegraph Equation

Advances in Mathematical Physics

2020

The fractional telegraph equation is a kind of important evolution equation, which has an important application in signal analysis such as transmission and propagation of electrical signals. However, it is difficult to obtain the corresponding analytical solution, so it is of great practical value to study the numerical solution. In this paper, the alternating segment pure explicit-implicit (PASE-I) and implicit-explicit (PASI-E) parallel difference schemes are constructed for time fractional telegraph equation. Based on the alternating segment technology, the PASE-I and PASI-E schemes are constructed of the classic explicit scheme and implicit scheme. It can be concluded that the schemes are unconditionally stable and convergent by theoretical analysis. The convergence order of the PASE-I and PASI-E methods is second order in spatial direction and 3-α order in temporal direction. The numerical results are in agreement with the theoretical analysis, which shows that the PASE-I and PASI-E schemes are superior to the classical implicit schemes in both accuracy and efficiency. This implies that the parallel difference schemes are efficient for solving the time fractional telegraph equation.

“…Lu et al (2015) [35] established a differential scheme for time fractional sub-diffusion equations and proposed a fast algorithm based on its special structure. Wang et al (2016) [36] studied the parallel algorithm of the implicit difference scheme for Caputo fractional reaction-diffusion equation. After parallelization, the computational efficiency of the original scheme is improved.…”

Section: Introductionmentioning

confidence: 99%

A new parallel difference algorithm based on improved alternating segment Crank–Nicolson scheme for time fractional reaction–diffusion equation

2019

Adv Differ Equ

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-α 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.