Riemann-Liouville型分数阶扩散方程的显–隐和隐–显差分方法及数值分析

Explicit-Implicit and Implicit-Explicit Difference Methods and Numerical Analysis for Riemann-Liouville Type Fractional Diffusion Equation

摘要: 针对描述慢扩散现象的Riemann-Liouville (R-L)型分数阶扩散方程，构造了求解该问题的一类显–隐和隐–显差分格式。它是利用显式格式快速计算和隐式格式无条件稳定的优点，按时间层交替使用古典显式格式和隐式格式而得。使用傅里叶方法分析可知该格式为无条件稳定且收敛的。数值试验结果与理论分析结果一致，表明显–隐和隐–显格式的计算精度和计算效率均优于经典隐式格式，证实本文显–隐和隐–显格式对求解R-L型分数阶慢扩散方程是有效的。

Abstract: For the Riemann-Liouville (R-L) type fractional differential equations describing sub-diffusion phenomena, a kind of explicit-implicit and implicit-explicit difference schemes for numerically solving this problem is constructed. It takes advantage of the fast computation of explicit formats and the unconditional stability of implicit formats, alternatingly applies the classical explicit format and the implicit format by time layer. Then, using the Fourier method analysis, the format is unconditionally stable and convergent. The numerical test results are consistent with the theoret-ical analysis results, and show that the computational efficiency of the explicit-implicit and implic-it-explicit formats is better than the classical implicit format. The explicit-implicit and implic-it-explicit methods are feasible to solve the R-L fractional diffusion equation.