High-order efficient numerical method for solving a generalized fractional Oldroyd-B fluid model
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High-order efficient numerical method for solving a generalized fractional Oldroyd-B fluid model Bo Yu1 Received: 12 July 2020 / Revised: 26 October 2020 / Accepted: 27 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract 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 τ min {3−γ ,2−α} + h 4 , where γ ∈ (1, 2) and α ∈ (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. Keywords Fractional Oldroyd-B fluid model · High-order numerical method · Stability · Convergence Mathematics Subject Classification 65M06 · 65M12 · 65M15
1 Introduction In the past few years, fractional calculus has been widely applied in different disciplines of science and engineering [8,12,14,15,22,27]. A large number of analytical techniques and numerical methods for solving various kinds of fractional differential equations have been investigated, for details one can refer to [1,10,11,16].
B 1
Bo Yu [email protected] School of Mathematics and Statistics, Shandong University, Weihai 264209, Shandong, China
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B. Yu
The generalized fractional Oldroyd-B fluid model is a special class of the nonNewtonian fluids. Due to its wide application, many scholars have been attracted into the analytic and numerical study of this kind of fluid model in recent years. For example, Qi and Xu [17] obtained the exact analytical solutions for Stokes’ first problem for a viscoelastic fluid with the generalized Oldroyd-B model. Khan et al. [9] constructed the exact solutions for the accelerated flows of a generalized Oldroyd-B fluid. Zheng et al. [30] presented the analysis for MHD flow of a generalized Oldroyd-B fluid with fractional derivative inducing by an accelerating plate. Riaz et al. [19] analyzed the analytic solutions of Oldroyd-B fluid with fractional derivatives in a circular duct. Zhang et al. [29] studied the MHD flow and heat transfer analysis of fractional Oldroyd-B nanofluid between two coaxial cylinders. Vasileva et al. [23] studied the alternating direction implicit schemes for the two-dimension
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