Fast difference scheme for a tempered fractional Burgers equation in porous media

Abstract

In this paper we present a fast difference scheme for a tempered fractional Burgers equation. The model arises in describing the wave propagation in porous media with the power law and exponential attenuation. To avoid solving the discrete algebraic system by iteration, we present a linearized difference approximation for the nonlinear term. To reduce the numerical computational cost, we propose a fast algorithm for tempered fractional derivative. The fast algorithm is based on the summation of exponentials technique which is developed in Jiang et al. (2017). The theoretical analysis shows that our fast difference scheme is unconditionally stable and convergence. Numerical results demonstrate a competitive performance of the proposed fast difference scheme in comparison to the direct difference scheme.