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In this work, two novel classes of structure-preserving spectral Galerkinmethods are proposed which based on the Crank-Nicolson scheme and theexponential scalar auxiliary variable method respectively, for solving thecoupled fractional nonlinear Klein-Gordon-Schr\"odinger equation. The paperfocuses on the theoretical analyses and computational efficiency of theproposed schemes, the Crank-Nicoloson scheme is proved to be unconditionallyconvergent and has the maximum-norm boundness of numerical solutions. Theexponential scalar auxiliary variable scheme is linearly implicit anddecoupled, but lack of the maximum-norm boundness, also, the energy structurehas been modified. Subsequently, the efficient implementations of the proposedschemes are introduced in detail. Both the theoretical analyses and thenumerical comparisons show that the proposed spectral Galerkin methods havehigh efficiency in long-time computations.

Unconditional convergence of conservative spectral Galerkin methods for the coupled fractional nonlinear Klein-Gordon-Schr\"odinger equations