Finite difference scheme for simulating a generalized two-dimensional multi-term time fractional non-Newtonian fluid model

Yanqin Liu,Libo Feng,Hongguang Sun,Xiuling Yin

doi:10.1186/s13662-018-1876-4

Abstract

A finite difference scheme, based upon the Crank–Nicolson scheme, is applied to the numerical approximation of a two-dimensional time fractional non-Newtonian fluid model. This model not only possesses a multi-term time derivative, but also contains a special time fractional operator on the spatial derivative. And a very important lemma is proposed and also proved, which plays a vital role in the proof of the unconditional stability. The stability and convergence of the finite difference scheme are discussed and theoretically proved by the energy method. Numerical experiments are given to validate the accuracy and efficiency of the scheme, and the results indicate that this Crank–Nicolson difference scheme is very effective for simulating the generalized non-Newtonian fluid diffusion model.

Highlights

1 Introduction Fractional partial differential equations have been applied to many anomalous phenomena and complex systems in natural science and engineering technology fields [1,2,3]
Numerical methods to solve fractional equations mainly are finite difference methods [32,33,34], finite element methods [35, 36], finite volume methods [37, 38], and spectral methods [39,40,41]. For these multiterm fractional fluid models, Bazhlekova et al [42] proposed a finite difference method to solve the viscoelastic flow with generalized fractional Oldroyd-B model, and they utilized the Grünwald–Letnikov formula to approximate the Riemann–Liouville time fractional derivative; the results were low accuracy and lacked theoretical analysis
Remark 1 When 0 < β < 0.4811, d2(β) – 2d1(β) + d0(β) < 0. Regarding these coefficients’ property, literature [44] proved that this Crank–Nicolson difference scheme is only conditionally stable

Summary

Introduction

Fractional partial differential equations have been applied to many anomalous phenomena and complex systems in natural science and engineering technology fields [1,2,3]. Daftardar–Gejji et al [13] obtained the solutions of multi-term time fractional diffusion wave equations under different homogeneous or non-homogeneous boundary conditions. Numerical methods to solve fractional equations mainly are finite difference methods [32,33,34], finite element methods [35, 36], finite volume methods [37, 38], and spectral methods [39,40,41] For these multiterm fractional fluid models, Bazhlekova et al [42] proposed a finite difference method to solve the viscoelastic flow with generalized fractional Oldroyd-B model, and they utilized the Grünwald–Letnikov formula to approximate the Riemann–Liouville time fractional derivative; the results were low accuracy and lacked theoretical analysis. We use finite difference method to solve the generalized twodimensional multi-term time fractional Oldroyd-B fluid equation. Inspired by the above works and motivated by potential applications, we will consider the following generalized two-dimensional multi-term time fractional non-Newtonian fluid diffusion equation: c1C0 Dγt u. We present a numerical example to demonstrate the effectiveness of our method and draw some conclusions

Preliminary

Conclusion