A numerically efficient Hamiltonian method for fractional wave equations

摘要

In this work, we consider a partial differential equation that extends the well known wave equation. The model under consideration is a multidimensional equation which includes the presence of both a damping term and a fractional Laplacian of the Riesz type. Homogeneous Dirichlet boundary conditions on a closed and bounded spatial interval are considered in this work. The mathematical model has a fractional Hamiltonian which is conserved when the damping coefficient is equal to zero, and dissipated otherwise. Motivated by these facts, we propose a finite-difference method to approximate the solutions of the continuous model. The method is an implicit scheme which is based on the use of fractional centered differences to approximate the spatial fractional derivatives of the model. A discretized form of the Hamiltoninan is also proposed in this work, and we prove analytically that the method is capable of preserving/dissipating the discrete energy when the continuous model preserves/dissipates the energy. We establish rigorously the properties of consistency, stability and convergence of the method, and provide some a priori bounds for the numerical solutions. Moreover, we prove the existence and the uniqueness of the numerical solutions as well as the unconditional stability of the method in the linear regime. Some computer simulations that assess the capability of the method to preserve/dissipate the energy are carried out for illustration purposes. (C) 2018 Elsevier Inc. All rights reserved.