On the solution of hyperbolic two-dimensional fractional systems via discrete variational schemes of high order of accuracy

In this work, we consider a general class of damped wave equations in two spatial dimensions. The model considers the presence of Weyl space-fractional derivatives as well as a generic nonlinear potential. The system has an associated positive energy functional when damping is not present, in which case, the model is capable of preserving the energy throughout time. Meanwhile, the energy of the system is dissipated in the damped scenario. In this work, the Weyl space-fractional derivatives are approximated through second-order accurate fractional centered differences. A high-order compact difference scheme with fourth order accuracy in space and second order in time is proposed. Some associated discrete quantities are introduced to estimate the energy functional. We prove that the numerical method is capable of conserving the discrete variational structure under the same conditions for which the continuous model is conservative. The positivity of the discrete energy of the system is also discussed. The properties of consistency, solvability, stability and convergence of the proposed method are rigorously proved. We provide some numerical simulations that illustrate the agreement between the physical properties of the continuous and the discrete models.