时间分数阶慢扩散方程的一类并行计算方法

A class of parallel computation method for time fractional sub-diffusion equation

针对时间分数阶慢扩散方程,提出一类并行差分方法——交替分段纯显-隐(pure alternative segment explicit-implicit,PASE-I)和交替分段纯隐-显(pure alternative segment implicit-explicit,PASI-E)差分方法。这类方法是将古典显式格式、古典隐式格式与交替分段技术相结合构造出的一类具有并行本性的差分方法。理论证明了PASE-I和PASI-E格式解的存在唯一性,采用傅里叶方法和数学归纳法证明了格式是无条件稳定且收敛的。数值试验表明:PASE-I格式和PASI-E格式具有明显的并行计算性质,为空间二阶、时间2-α阶收敛,并且在计算效率上相比串行的隐式格式有大幅度提高,本方法求解时间分数阶慢扩散方程是可行的。

A kind of parallel numerical method with parallel nature which are the pure alternative segment explicit-implicit（PASEI）and implicit-explicit（PASI-E）difference method is constructed for solving the time fractional diffusion equation.It is a combination of the explicit scheme,the implicit scheme and the alternating segment technique.Theoretical analysis has shown that the solutions of PASE-I and PASI-E scheme not only exist but also unique.At the same time the stability and convergence of the schemes are proved by the Fourier method and the mathematical induction.Numerical experiments indicate that the theoretical analysis which showed that the convergence rates are spatially second-order and temporally 2-αorder.Meanwhile the PASE-I and PASI-E schemes have obvious parallel properties due to the higher computational efficiency.Therefore it is feasible to use the scheme to solve the time fractional diffusion equation.