High-order efficient numerical method for solving a generalized fractional Oldroyd-B fluid model

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 Oτmin{3-γ,2-α}+h4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\left( \tau ^{\min {\{3-\gamma ,2-\alpha }\}}+h^{4}\right) $$\end{document}, where γ∈(1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (1,2)$$\end{document} and α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} 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.