Abstract This paper is concerned with a numerical method for a class of one-dimensional multi-term time-fractional convection-reaction-diffusion problems, where the differential equation contains a sum of the Caputo time-fractional derivatives of different orders between 0 and 1. In general the solutions of such problems typically exhibit a weak singularity at the initial time. A compact exponential finite difference method, using the well-known L1 formula for each time-fractional derivative and a fourth-order compact exponential difference approximation for the spatial discretization, is proposed on a mesh that is generally nonuniform in time and uniform in space. Taking into account the initial weak singularity of the solution, the stability and convergence of the method is proved and the optimal error estimate in the discrete L2-norm is obtained by developing a discrete energy analysis technique which enables us to overcome the difficulties caused by the nonsymmetric discretization matrices. The error estimate shows that the method has the spatial fourth-order convergence, and reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence. The extension of the method to two-dimensional problems is also discussed. Numerical results confirm the theoretical convergence result, and show the applicability of the method to convection dominated problems.

中文翻译：

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具有非光滑解的多项时间分数阶对流-反应-扩散问题的紧凑指数差分法

摘要 本文涉及一类一维多项时间分数阶对流-反应-扩散问题的数值方法，其中微分方程包含 0 到 1 之间不同阶数的 Caputo 时间分数阶导数之和。 . 一般而言，此类问题的解在初始时通常表现出弱奇异性。在时间上通常不均匀且空间上均匀的网格上，提出了一种紧致指数有限差分方法，该方法对每个时间分数阶导数使用众所周知的 L1 公式和用于空间离散化的四阶紧致指数差分近似。考虑到解的初始弱奇异性，证明了该方法的稳定性和收敛性，并通过开发离散能量分析技术获得了离散 L2 范数中的最优误差估计，该技术使我们能够克服非对称离散矩阵带来的困难。误差估计表明该方法具有空间四阶收敛性，并揭示了如何选择合适的网格参数以获得时间最优收敛性。还讨论了该方法对二维问题的扩展。数值结果证实了理论收敛结果，并表明该方法对以对流为主的问题的适用性。误差估计表明该方法具有空间四阶收敛性，并揭示了如何选择合适的网格参数以获得时间最优收敛性。还讨论了该方法对二维问题的扩展。数值结果证实了理论收敛结果，并表明该方法对以对流为主的问题的适用性。误差估计表明该方法具有空间四阶收敛性，并揭示了如何选择合适的网格参数以获得时间最优收敛性。还讨论了该方法对二维问题的扩展。数值结果证实了理论收敛结果，并表明该方法对以对流为主的问题的适用性。