In this paper, we propose a fast high order method for the time-fractional diffusion equation on unbounded spatial domains. The proposed numerical method is a combination of a time-stepping scheme and spectral method for the spatial discretization. First, we reformulate the unbounded domain problem into a bounded domain problem by introducing suitable artificial boundary conditions. Then the time fractional derivatives involved in the equation and the artificial boundary condition are discretized using the so-called L2 formula and sum-of-exponentials (SOE) approximation. The former has been a popular formula for discretization of the Caputo fractional derivative, while the latter is a computational cost reducing technique frequently employed in recent years for convolution integrals. The spatial discretization makes use of the standard Legendre spectral method. The stability and the accuracy of the full discrete problem are analyzed. Our obtained theoretical results include a rigorous proof of the convergence order for both uniform mesh and graded mesh, and a stability proof for the uniform mesh. Finally, several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.

在本文中，我们提出了一种用于无界空间域上的时间分数扩散方程的快速高阶方法。所提出的数值方法是时间步进方案和空间离散化谱方法的组合。首先，我们通过引入合适的人工边界条件将无界域问题重新表述为有界域问题。然后使用所谓的 L2 公式和指数和 (SOE) 近似对方程中涉及的时间分数阶导数和人工边界条件进行离散化。前者是 Caputo 分数阶导数离散化的流行公式，而后者是近年来卷积积分中经常使用的一种降低计算成本的技术。空间离散化利用标准勒让德谱方法。分析了全离散问题的稳定性和精度。我们获得的理论结果包括均匀网格和分级网格收敛阶的严格证明，以及均匀网格的稳定性证明。最后，提供了几个数值例子来验证理论结果并证明该方法的有效性。