时间分数阶奇异摄动对流扩散方程参数一致的有限差分方法

A Parameter-Uniform Finite Difference Method for a Time-Fractional Singularly Perturbed Convection-Diffusion Equation

该文讨论了一类时间分数阶奇异摄动对流扩散方程的数值方法.首先在均匀网格下,利用L1近似方法对时间导数进行离散,证明了该方法的稳定性并给出了局部截断误差的估计.然后基于Vulanovic-Bakhvalov网格,利用迎风有限差分方法对空间导数进行离散.接着运用闸函数法和离散算子的极大值原理证明了全离散格式的收敛性.最后通过数值实验验证了理论结果.

In this paper a finite difference method for time-fractional singularly perturbed convection-diffusion problems is concerned. First, we discretize the time derivative on a uniform grid by the L1 approximation method, and we prove the stability of this method and give an estimation for the local truncation error. Second, based on the Vulanovic-Bakhvalov grid, we discretize the spatial derivative by the upwind finite difference method. Then we prove the convergence of the fully discrete scheme by the barrier function method and the maximum principle of the discrete operator. Finally the theoretical results are verified in the numerical experiment.