Structure-preserving numerical methods for the two dimensional nonlinear fractional wave equation

Abstract

This paper introduces the structure-preserving numerical methods for the two dimensional nonlinear fractional wave equation. By using the variational principle with fractional Laplace, the equation is transformed into a Hamiltonian system with symplectic and multi-symplectic structure, and they show the corresponding conservation laws. Then a numerical method is proposed with the Fourier pseudospectral method in space and midpoint method in time. It is proved that the proposed numerical method preserves the corresponding conservation laws in the discrete sense. Furthermore, one investigates energy errors of fully discrete schemes, and discusses convergence of the proposed schemes which are second-order accuracy in time and spectral accuracy in space. Finally, the validity and accuracy of the theoretical results are verified by several numerical examples.