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Structure-preserving algorithms for the two-dimensional fractional Klein-Gordon-Schrödinger equation. (English) Zbl 1442.65219
Summary: This paper aims to construct structure-preserving numerical schemes for the two-dimensional space fractional Klein-Gordon-Schrödinger equation, which are based on the newly developed partitioned averaged vector field methods. First, we derive an equivalent equation, and reformulate the equation as an infinite dimensional canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian. Then, we use the Fourier pseudo-spectral method to discrete the equation in space direction and obtain a semi-discrete conservative system, which can be reformulated a finite dimensional canonical Hamiltonian system. Further applying the partitioned averaged vector field methods to the semi-discrete system gives a class of fully-discrete schemes that can preserve the mass and energy exactly. Numerical examples are provided to confirm our theoretical analysis results at last.

##### MSC:
 65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs 65P10 Numerical methods for Hamiltonian systems including symplectic integrators 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 26A33 Fractional derivatives and integrals 35R11 Fractional partial differential equations 35Q55 NLS equations (nonlinear Schrödinger equations)
FODE
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