A backward Euler alternating direction implicit difference scheme for the three‐dimensional fractional evolution equation
A backward Euler alternating direction implicit difference scheme for the three‐dimensional fractional evolution equation
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DOI：
10.1002/num.22239
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发表时间：
2018-05
影响因子：
3.9
通讯作者：
Hongbin Chen;Da Xu;Jiliang Cao;Junjun Zhou
中科院分区：
文献类型：
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作者：
Hongbin Chen;Da Xu;Jiliang Cao;Junjun Zhou
对三维分数阶发展方程建立了一种后向Euler交替方向隐式（ADI）差分格式，并进行了数值分析.在我们的方法中，Riemann-Liouville分数阶积分项通过Lubich建议的一阶卷积求积来处理。同时，采用ADI技术将多维问题简化为一系列一维问题。采用有限差分法进行空间离散，构造了一个全离散差分格式。定义了两个新的内积和相应的范数来分析该方案。稳定性和收敛性的验证是基于与卷积求积相关的真实的二次型的非负特性。数值实验证明了我们的方案的有效性。
A backward Euler alternating direction implicit (ADI) difference scheme is formulated and analyzed for the three‐dimensional fractional evolution equation. In our method, the Riemann‐Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, an ADI technique is adopted to reduce the multidimensional problem to a series of one‐dimensional problems. A fully discrete difference scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. Numerical experiments are reported to demonstrate the efficiency of our scheme.