Two efficient compact ADI methods for the two-dimensional fractional Oldroyd-B model

Abstract

The objective of this paper is to present efficient numerical algorithms to resolve the two-dimensional fractional Oldroyd-B model. Firstly, two compact alternating direction implicit (ADI) methods are constructed with convergence orders $O({\tau }^{\min \{3-\gamma ,2-\beta ,1+\gamma -2\beta \}}+h_x^4+h_y^4)$ and $O({\tau }^{\min \{3-\gamma ,2-\beta \}}+h_x^4+h_y^4)$, where γ and β are orders of two Caputo fractional derivatives, τ, $h_x$ and $h_y$ are the time and space step sizes, respectively. Secondly, the convergence analyses of the proposed compact ADI methods are investigated strictly utilizing the energy estimation technique. Lastly, the two compact ADI methods are implemented to confirm the effectiveness of the convergence analysis. The convergence orders of the two compact ADI methods are separately tested in the direction of time and space, the CPU times are computed compared with the direct compact scheme to demonstrate the efficiency of the derived compact ADI methods, numerical results are also compared with the existing literature. All the numerical simulation results are listed in tabular forms which manifest the validity of the derived compact ADI methods.