非线性变阶空间-时间分数阶对流-扩散方程的全隐式有限差分格式

Fully Implicit Finite Difference Scheme for the Nonlinear Variable-Order Space-Time Fractional Advection-Diffusion Equation

针对非线性变阶空间-时间分数阶对流-扩散方程的初边值问题,提出一种全隐式有限差分格式.首先,分别对Riemann-Liouville型变时间分数阶导数算子和Riemann-Liouville型变空间分数阶导数算子和广义Riesz分数阶导数算子进行离散化处理;然后,通过离散的能量方法证明全隐式有限差分格式的稳定性和收敛性,并验证其收敛阶为O(τ+h);最后,通过数值算例检验该方法.试验结果表明:全隐式有限差分格式求解非线性变阶空间-时间分数阶对流-扩散方程初边值问题是可行和有效的.

A fully implicit finite difference scheme for the nonlinear variable-order fractional advection-diffusion equation was considered.Firstly,the Riemann-Liouville variable order time fractional derivative,the Riemann-Liouville variable order space fractional derivative and the generalized Riesz fractional derivative were discretized respectively.Then,the convergence and the stability of the fully implicit finite difference scheme were obtained by discrete energy method,and the convergence order of the scheme was O(τ+h).Finally,a numerical example was provided to test this method.The results demonstrated the feasibility and the efficiency of the proposed method.