A Compact Difference Scheme for Time-Space Fractional Nonlinear Diffusion-Wave Equations with Initial Singularity (Advances in Applied Mathematics and Mechanics, 2023)
In this paper, we present a linearized compact difference scheme for one-dimensional time-space fractional nonlinear diffusion-wave equations with initialboundary value conditions. The initial singularity of the solution is considered, whichoften generates a singular source and increases the difficulty of numerically solvingthe equation. The Crank-Nicolson technique, combined with the midpoint formulaand the second-order convolution quadrature formula, is used for the time discretization. To increase the spatial accuracy, a fourth-order compact difference approximation, which is constructed by two compact difference operators, is adopted for spatial discretization. Then, the unconditional stability and convergence of the proposedscheme are strictly established with superlinear convergence accuracy in time andfourth-order accuracy in space. Finally, numerical experiments are given to supportour theoretical results.