求解时间分布阶扩散方程的两个高阶有限差分格式

Two High-Order Difference Schemes for Solving Time Distributed-Order Diffusion Equations

基于复化Simpson公式和复化两点Gauss-Legendre公式,构造了两个求解时间分布阶扩散方程的高阶有限差分格式.不同于以往文献中提出的时间一阶或二阶格式,这两种格式在时间方向都具有三阶精度,而在分布阶和空间方向可达到四阶精度.数值结果表明,两种算法都是稳定且收敛的,从而是有效的.两种格式的收敛速率也通过数值实验进行了验证,并且通过和文献中的算法对比可以得出其更为高效.

Based on the composite Simpson’ s quadrature rule and the composite 2-point Gauss-Legendre quadrature rule, 2 high-order finite difference schemes were proposed for solving time distributed-order diffusion equations. Other than the existing methods whose convergence rates are only 1st-order or 2nd-order in the temporal domain, the proposed 2 schemes both have 3rd-order convergence rates in the temporal domain, and 4th-order rates in the spatial domain and the distributed order, respectively. Such high-order convergence rates were further verified with numerical examples. The results show that, both of the proposed 2 schemes are stable, and have higher accuracy and efficiency compared with existing algorithms.