An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations
An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations
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DOI：
10.1016/j.matcom.2019.05.017
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发表时间：
2019-12
期刊：
影响因子：
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通讯作者：
W. Qiu;Hongbin Chen;Xuan Zheng
中科院分区：
文献类型：
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作者：
W. Qiu;Hongbin Chen;Xuan Zheng
考虑一维时间分数阶Burgers方程的时间截断为2− α（0< α< 1），空间截断为2阶的隐式差分格式.采用Caputo意义下分数阶导数的L1-离散公式.二阶空间导数采用三点中心公式近似，非线性对流项采用基于分段线性检验函数的Galerkin方法离散。利用能量法证明了该算法在L∞范数下的稳定性和收敛性.同时，提出并实现了一种新的迭代算法来求解非线性系统。数值实验表明，本文的结果与理论分析一致，并与已有的迭代算法进行了比较，验证了本文方法的有效性.
An implicit difference scheme with the truncation of order 2− α (0< α< 1) for time and order 2 for space is considered for the one-dimensional time-fractional Burgers equations. The L 1-discretization formula of the fractional derivative in the Caputo sense is employed. The second-order spatial derivative is approximated by means of the three-point centered formula and the nonlinear convection term is discretized by the Galerkin method based on piecewise linear test functions. The stability and convergence in the L∞ norm are proved by the energy method. Meanwhile, a novel iterative algorithm is proposed and implemented to solve the nonlinear systems. Numerical experiment shows that the results are consistent with our theoretical analysis, and the comparison between the proposed iterative algorithm and the existing methods shows the efficiency of our method.