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

TL;DR

This paper introduces two structure-preserving spectral Galerkin methods for the coupled fractional nonlinear Klein-Gordon-Schrödinger equations, demonstrating their theoretical convergence and computational efficiency for long-term simulations.

Contribution

It proposes two novel spectral Galerkin schemes, one unconditionally convergent with maximum-norm bounds and the other linearly implicit and decoupled, enhancing long-time computational efficiency.

Findings

Crank-Nicolson scheme is unconditionally convergent.

Maximum-norm boundness of numerical solutions is proved.

Both methods show high efficiency in long-time simulations.

Abstract

In this work, two novel classes of structure-preserving spectral Galerkin methods are proposed which based on the Crank-Nicolson scheme and the exponential scalar auxiliary variable method respectively, for solving the coupled fractional nonlinear Klein-Gordon-Schr\"odinger equation. The paper focuses on the theoretical analyses and computational efficiency of the proposed schemes, the Crank-Nicoloson scheme is proved to be unconditionally convergent and has the maximum-norm boundness of numerical solutions. The exponential scalar auxiliary variable scheme is linearly implicit and decoupled, but lack of the maximum-norm boundness, also, the energy structure has been modified. Subsequently, the efficient implementations of the proposed schemes are introduced in detail. Both the theoretical analyses and the numerical comparisons show that the proposed spectral Galerkin methods have high efficiency in long-time computations.