A novel fast second order approach with high-order compact difference scheme and its analysis for the tempered fractional Burgers equation

This research focuses on devising a new fast difference scheme to simulate the Caputo tempered fractional derivative (TFD). We introduce a fast tempered ${}^{\lambda }F\pounds 2-1\sigma$ difference method featuring second-order precision for a tempered time fractional Burgers equation (TFBE) with tempered parameter $\lambda$ and fractional derivative of order $\alpha$ ($0<\alpha <1$). The model emerges in characterizing the propagation of waves in porous material with the power law kernel and exponential attenuation. To circumvent iteratively resolving the discretized algebraic system, we introduce a linearized difference operator for approximating the nonlinear terms appearing in the model. The second-order fast tempered scheme relies on the sum of exponents (SOE) technique. The method’s convergence and stability are analyzed theoretically, establishing unconditional stability and maintaining the accuracy of order $O({\tau }^2+h^2+ϵ)$, where $\tau$ denotes the temporal step size, $ϵ$ is the tolerance error and $h$ is the spatial step size. Moreover, a novel compact finite difference (CFD) scheme of high order is developed for tempered TFBE. We investigate the stability and convergence of this fourth-order compact scheme utilizing the energy method. Numerical simulations indicate convergence to $O({\tau }^2+h^4+ϵ)$ under robust regularity assumptions. Our computational results align with theoretical analysis, demonstrating good accuracy while reducing computational complexity and storage needs compared to the standard tempered ${}^{\lambda }\pounds 2-1\sigma$ scheme, with significant reduction in CPU time. Numerical outcomes showcase the competitive performance of the fast tempered ${}^{\lambda }F\pounds 2-1\sigma$ scheme relative to the standard ${}^{\lambda }\pounds 2-1\sigma$.