An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements
An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements
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DOI：
10.1093/imanum/drp057
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发表时间：
2011-04
影响因子：
2.1
通讯作者：
K. Mustapha
中科院分区：
文献类型：
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作者：
K. Mustapha
研究了一类参数取值范围为−1<；α<；0的次扩散方程的数值解。对于时间离散化，我们使用隐式有限差分Crank-Nicolson方法，证明了误差为k2+α阶，其中k表示最大时间步长。采用非均匀时间步长来补偿t=0时精确解的奇异行为。我们还考虑了将空间线性有限元应用于所提出的时间步进格式而得到的全离散格式。我们证明了附加误差为h2max(1，logk−1)，其中h是空间网格的参数。对一些样本问题的数值实验验证了我们的理论结果。
The numerical solution for a class of sub-diffusion equations involving a parameter in the range −1 < α < 0 is studied. For the time discretization, we use an implicit finite-difference Crank–Nicolson method and show that the error is of order k2+α , where k denotes the maximum time step. A nonuniform time step is employed to compensate for the singular behaviour of the exact solution at t = 0. We also consider a fully discrete scheme obtained by applying linear finite elements in space to the proposed time-stepping scheme. We prove that the additional error is of order h2 max(1, log k−1), where h is the parameter for the space mesh. Numerical experiments on some sample problems demonstrate our theoretical result.