This paper investigates the high-order efficient numerical method with the corresponding stability and convergence analysis for the generalized fractional Oldroyd-B fluid model. Firstly, a high-order compact finite difference scheme is derived with accuracy \(O\left( \tau ^{\min {\{3-\gamma ,2-\alpha }\}}+h^{4}\right) \), where \(\gamma \in (1,2)\) and \(\alpha \in (0,1)\) are the orders of the time fractional derivatives. Then, by means of a new inner product, the unconditional stability and convergence in the maximum norm of the derived high-order numerical method have been discussed rigorously using the energy method. Finally, numerical experiments are presented to test the convergence order in the temporal and spatial direction, respectively. To precisely demonstrate the computational efficiency of the derived high-order numerical method, the maximum norm error and the CPU time are measured in contrast with the second-order finite difference scheme for the same temporal grid size. Additionally, the derived high-order numerical method has been applied to solve and analyze the flow problem of an incompressible Oldroyd-B fluid with fractional derivative model bounded by two infinite parallel rigid plates.

本文研究了广义分式Oldroyd-B流体模型的高阶有效数值方法，以及相应的稳定性和收敛性分析。首先，导出具有精度\（O \ left（\ tau ^ {\ min {\ {3- \ gamma，2- \ alpha} \}} + h ^ {4} \ right的高阶紧致有限差分格式）\），其中\（\ gamma \ in（1,2）\）和\（\ alpha \ in（0,1）\）是时间分数导数的阶数。然后，借助于新的内积，使用能量方法严格讨论了导出的高阶数值方法的最大范数的无条件稳定性和收敛性。最后，通过数值实验分别测试了时间和空间方向的收敛顺序。为了精确地证明派生的高阶数值方法的计算效率，与相同时间网格大小的二阶有限差分方案相比，测量了最大范数误差和CPU时间。此外，导出的高阶数值方法已被用于解决和分析不可压缩的Oldroyd-B流体的流动问题，其分数导数模型由两个无限平行的刚性板界定。