Fast dissipation-preserving difference scheme for nonlinear generalized wave equations with the integral fractional Laplacian

Abstract

In this paper, we construct a dissipation-preserving difference scheme for two-dimensional nonlinear generalized wave equations with the integral fractional Laplacian. We discuss the discrete dissipation property of the scheme in detail, and we also analyze the existence, uniqueness and the unconditional convergence of the proposed scheme. Further, we reveal that the spatial discretization generates a block-Toeplitz coefficient matrix, and it will be ill-conditioned as the spatial grid mesh number M and the fractional order α increase. Thus, we exploit an efficient linearized iteration algorithm for the nonlinear system, such that it can be efficiently solved by the Krylov subspace solver with a suitable preconditioner, where the two-dimensional fast Fourier transform is used in the solver to accelerate the matrix-vector product. Extensive numerical experiments are provided to verify the theoretical analysis and discrete dissipation law of the proposed scheme in long-time computations.