Efficient two-dimensional simulations of the fractional Szabo equation with different time-stepping schemes

The modified Szabo wave equation is one of the various models that have been developed to model the power law frequency-dependent attenuation phenomena in lossy media. The purpose of this study is to develop two different efficient numerical methods for the two-dimensional Szabo equation and to compare the relative merits of each method. In both methods we employ the ADI scheme to split directions, however, we use different time discretization. Specifically, in the first ADI method (ADI-I) we include a third-order correction term to achieve second-order convergence for smooth solutions, hence extending the work of Sun and Wu (2006). In the second ADI method (ADI-II), we employ the scheme in Zeng et al. (submitted for publication) to two dimensional fractional wave equation using multiple correction terms to enhance accuracy for non-smooth solutions. Our simulation results show that both methods are computationally efficient for the fractional wave equation but have different advantages in terms of accuracy. Specifically, ADI-II seems to produce more accurate results than ADI-I for non-smooth solutions. However, for smooth solutions and fractional order close to two, ADI-I seems to outperform ADI-II.