Abstract This paper is concerned with numerical methods for a class of nonhomogeneous Neumann problems of time-fractional reaction–diffusion equations with variable coefficients. The solutions of this kind of problems often have weak singularity at the initial time. This makes the existing numerical methods with uniform time mesh often lose accuracy. In this paper, we propose and analyze a high-order compact finite difference method with nonuniform time mesh. The time-fractional derivative is approximated by Alikhanov’s high-order approximation on a class of fitted time meshes. For the spatial variable coefficient differential operator, a new fourth-order boundary discretization is developed under the nonhomogeneous Neumann boundary condition, and then a new fourth-order compact finite difference approximation on a space uniform mesh is obtained. Under the assumption of the weak initial singularity of solution, we prove that for the general case of the variable coefficients, the proposed method is unconditionally stable and the numerical solution converges to the solution of the problem under consideration. The convergence result also gives an optimal error estimate of the numerical solution in the discrete L 2 -norm, which shows that the method has the spatial fourth-order convergence, while it attains the temporal optimal second-order convergence provided a proper mesh grading parameter is employed. Numerical results that confirm the sharpness of the error analysis are presented.

中文翻译：

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变系数时分反应扩散方程 Neumann 问题拟合网格的高阶紧致差分法

摘要 本文研究一类变系数时间分数阶反应扩散方程的非齐次Neumann问题的数值方法。这类问题的解在初始时往往具有弱奇异性。这使得现有的具有均匀时间网格的数值方法经常失去精度。在本文中，我们提出并分析了一种具有非均匀时间网格的高阶紧致有限差分方法。时间分数导数通过 Alikhanov 对一类拟合时间网格的高阶近似来近似。对于空间变系数微分算子，在非齐次Neumann边界条件下进行了新的四阶边界离散化，得到了空间均匀网格上的新的四阶紧致有限差分近似。在解的弱初始奇异性假设下，证明了对于变系数的一般情况，所提方法是无条件稳定的，数值解收敛于所考虑问题的解。收敛结果还给出了离散L 2 -范数下数值解的最优误差估计，表明该方法具有空间四阶收敛性，同时在提供适当的网格分级参数的情况下，它获得了时间最优二阶收敛性受雇。数值结果证实了误差分析的锐度。所提出的方法是无条件稳定的，数值解收敛于所考虑问题的解。收敛结果还给出了离散L 2 -范数下数值解的最优误差估计，表明该方法具有空间四阶收敛性，同时在提供适当的网格分级参数的情况下，它获得了时间最优二阶收敛性受雇。数值结果证实了误差分析的锐度。所提出的方法是无条件稳定的，数值解收敛于所考虑问题的解。收敛结果还给出了离散L 2 -范数下数值解的最优误差估计，表明该方法具有空间四阶收敛性，同时在提供适当的网格分级参数的情况下，它获得了时间最优二阶收敛性受雇。数值结果证实了误差分析的锐度。如果采用适当的网格分级参数，则它可以达到时间上的最佳二阶收敛。数值结果证实了误差分析的锐度。如果采用适当的网格分级参数，则它可以达到时间上的最佳二阶收敛。数值结果证实了误差分析的锐度。