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A high-order compact difference method on fitted meshes for Neumann problems of time-fractional reaction-diffusion equations with variable coefficients

机译:具有变系数的拟合网眼拟合网眼的高阶小差差法,变量系数的近距离反应 - 扩散方程存在

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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.
机译:本文涉及具有可变系数的一类时间分数反应扩散方程的一类非均匀Neumann问题的数值方法。这种问题的解决方案通常在初始时间具有弱奇异性。这使得具有均匀时间网格的现有数值方法经常损失精度。在本文中,我们提出了一种具有非均匀时间网的高阶紧凑型有限差分方法。时间分数导数由Alikhanov在一类拟合时间网上的高阶近似值近似。对于空间变量系数差分运算符,在非均匀Neumann边界条件下开发了新的四阶边界离散化,然后获得空间均匀网格上的新的四阶紧凑型有限差分近似。在解决方案的弱初始奇异性的假设下,我们证明,对于变系数的一般情况,所提出的方法是无条件的稳定性,数值溶液会聚到所考虑的问题的解决方案。收敛结果还给出了离散L〜2-NOM中的数字解决方案的最佳误差估计,这表明该方法具有空间四阶收敛,而它达到了时间最佳的二阶收敛,提供了适当的网格分级参数被采用。呈现了确认误差分析锐度的数值结果。

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