The Hamiltonian structure and fast energy-preserving algorithms for the fractional Klein-Gordon equation

Abstract

In this paper, an energy-preserving difference scheme is proposed for solving the space fractional Klein-Gordon equation based on the Hamiltonian form of the equation. First, we study some properties of the fractional Laplacian operator and reformulate the equation as an infinite-dimension canonical Hamiltonian system. Then, we use the fractional centered difference formula to discrete the equation and derive a semi-discrete Hamiltonian system that can conserve the semi-discrete energy. Subsequently, a fully-discrete scheme is obtained by applying the averaged vector field method to the semi-discrete system. The resulting scheme's unconditional point-wise error estimate is discussed by using the “cut-off” technique. Furthermore, a fast solver is presented to reduce the computational complexity of the scheme based on the fast Fourier transformation technique in practical computation. Finally, some numerical experiments are displayed to demonstrate the efficiency and conservation of the constructed scheme in long time simulation.