Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient
Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient
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DOI：
10.1016/j.apm.2013.10.037
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发表时间：
2014-08
影响因子：
5
通讯作者：
Xuan Zhao;Qinwu Xu
中科院分区：
文献类型：
--
作者：
Xuan Zhao;Qinwu Xu
本文考虑变系数时间分数阶次扩散方程在Dirichlet边界条件和Neumann边界条件下的数值解。提出了求解Dirichlet边界条件方程的紧致差分格式。借助于新引入的变系数范数，严格证明了格式在最大范数下的无条件稳定性和全局收敛。收敛阶为O(τ2-α+h4)，其中τ为时间网格大小，α为分数导数阶，h为空间网格大小。此外，在Neumann边界条件下，通过引入新的中间变量，得到了一个盒型格式。给出了箱型格式在极大范数下的稳定性和全局收敛性质。数值实验验证了所提方案的理论结果。
In this paper, we consider the numerical solutions of the time fractional sub-diffusion equation with the variable coefficient subject to both Dirichlet boundary conditions and Neumann boundary conditions. A compact difference scheme is proposed for solving the equation with Dirichlet boundary conditions. The unconditional stability and the global convergence of the scheme in the maximum norm are proved rigorously with the help of the newly introduced norms regarding to the variable coefficient. The convergence order is O (τ 2-α+ h 4), where τ is the temporal grid size, α is the order of fractional derivative and h is the spatial grid size. Besides, a box-type scheme is derived by introducing new intermediate variable for the problem with Neumann boundary conditions. The stability and the global convergence of box-type scheme in maximum norm are also presented. Numerical experiments are carried out to confirm the theoretical results of the proposed schemes.