Energy dissipation-preserving GSAV-Fourier–Galerkin spectral schemes for space-fractional nonlinear wave equations in multiple dimensions

Abstract

In this paper, we utilize the generalized scalar auxiliary variable (GSAV) approach proposed in recent paper [SIAM J. Numer. Anal., 60 (2022), 1905–1931] for space-fractional nonlinear wave equation to construct a novel class of linearly implicit energy dissipation-preserving finite difference/spectral scheme. The unconditional energy dissipation property and unique solvability of the fully discrete scheme are first established. Next, we apply the mathematical induction to discuss the convergence results of the proposed scheme in one- and two-dimensional spaces without the assumption of global Lipschitz condition for the nonlinear term which is necessary for the almost all previous works. Moreover, the convergence of one-dimensional space is unconditional but conditional for two-dimensional space, due to the fractional Sobolev inequalities of one-dimensional space are not equivalent to the high-dimensional versions. Subsequently, the efficient implementations of the proposed schemes are introduced in detail. Finally, extensive numerical comparisons are reported to confirm the effectiveness of the proposed schemes and the correctness of the theoretical analyses.