A numerical method for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations

Abstract

Abstract Recently, numerous numerical schemes have been developed for solving single-term time-space fractional diffusion-wave equations. Among them, some popular methods were constructed by using the graded meshes due to the solution with low temporal regularity. In this paper, we present an efficient alternating direction implicit (ADI) scheme for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations. Firstly, the considered problem is equivalently transformed into its partial integro-differential form with the Riemann-Liouville integral and multi-term Caputo derivatives. Secondly, the ADI scheme is constructed by using the first-order approximations and L1 approximations to approximate the terms in time, and using the fractional centered differences to discretize the multi-term Riesz fractional derivatives in space. Furthermore, the fast implement of the proposed ADI scheme is discussed by the sum-of-exponentials technique for both Caputo derivatives and Riemann-Liouville integrals. Then, the solvability, unconditional stability and convergence of the proposed ADI scheme are strictly established. Finally, two numerical examples are given to support our theoretical results, and demonstrate the computational performances of the fast ADI scheme.