Fast high-order linearized exponential methods for efficient simulation of 2D time-fractional Burgers equation in polymer solution dynamics

Abstract

This study focuses on crafting and examining the high-order numerical technique for the two-dimensional time-fractional Burgers equation(2D-TFBE) arising in modelling of polymer solution. The time derivative of order \({\alpha }\) in the equation (where \(\alpha \in (0,1)\)) is approximated using the fast scheme, while space derivatives are discretized via a tailored finite point formula (TFPF) which relies on exponential basis. This method uses exponential functions to simultaneously fit the local solution’s properties in time and space, serving as basis functions within the TFPF framework. The analysis of convergence and stability of the method are rigorously examined theoretically and these are supported by the numerical examples showcasing its applicability and accuracy. It is proven that the method is unconditionally stable and maintains an accuracy of order \({\mathcal {O}}(\tau ^2+h_{\varkappa }+h_y+\epsilon + \varepsilon ^{-2}e^{-\frac{\beta _{n,m}^{k+1}}{2\varepsilon ^2}}+e^{-\gamma _0\frac{h}{\varepsilon }} )\), where \(\tau \) represents the temporal step size, and \(h_{\varkappa }\) and \(h_y\) are spatial step sizes. Computational outcomes align well with the theoretical analysis. Furthermore, when compared to the standard scheme, our method attains the same level of accuracy with significantly lowering computational demands and minimizing storage requirements. This proposed numerical scheme has higher convergence rate and significantly minimizes consumed CPU time compared to other methods.