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

Abstract

This research focuses on devising a new fast difference scheme to simulate the Caputo tempered fractional derivative (TFD). We introduce a fast tempered λF£2−1σ difference method featuring second-order precision for a tempered time fractional Burgers equation (TFBE) with tempered parameter λ and fractional derivative of order α (0<α<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(τ2+h2+ϵ), where τ 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(τ2+h4+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 λ£2−1σ scheme, with significant reduction in CPU time. Numerical outcomes showcase the competitive performance of the fast tempered λF£2−1σ scheme relative to the standard λ£2−1σ. © 2024 International Association for Mathematics and Computers in Simulation (IMACS)