A fully explicit variational integrator for multidimensional systems of coupled nonlinear fractional hyperbolic equations

摘要

In this work, we investigate a multidimensional system consisting of a finite (though arbitrary) number of coupled hyperbolic partial differential equations with fractional diffusion, constant damping and inertial times, and nonlinear reaction terms. Under suitable analytical conditions, the model has conserved quantities which are preserved in the absence of damping. We establish rigorously the conservation of the proposed quantities and, assuming that solutions of the model exist, we prove their boundedness. Motivated by these facts, we propose a finite-difference methodology to approximate the solutions of the continuous system. As its continuous counterpart, the discrete model has associated discrete quantities that estimate the Hamiltonian functional. Moreover, these quantities are preserved in the absence of damping, and they are dissipated when damping is present. To prove this feature of our finite-difference scheme, a new approximation form of the nonlinear reaction terms is proposed. This approach allows for the scheme to mimic the properties of the continuous system. The numerical properties of consistency, stability, boundedness and convergence of the scheme are proved rigorously. Some illustrative simulations confirm that the scheme is capable of preserving or dissipating the quantities, in agreement with the analytical results.