Abstract
In this work, we investigate numerically a one-dimensional wave equation in generalized form. The system considers the presence of constant damping and functional anomalous diffusion of the Riesz type. Reaction terms are also considered, in such way that the mathematical model can be presented in variational form when damping is not present. As opposed to previous efforts available in the literature, the reaction terms are not only functions of the solution. Instead, we consider the presence of smooth functions that depend on fractional derivatives of the solution function. Using a finite-difference approach, we propose a numerical scheme to approximate the solutions of the fractional wave equation. Along with this integrator, we propose discrete forms of the local and the total energy operators. In a first stage, we show rigorously that the energy properties of the continuous system are mimicked by our discrete methodology. In particular, we prove that the discrete system is dissipative (respectively, conservative) when damping is present (respectively, absent), in agreement with the continuous model. The theoretical numerical analysis of this system is more complicated in light of the presence of the functional form of the anomalous diffusion. To solve this problem, some novel technical lemmas are proved and used to establish the stability and the quadratic convergence of the scheme. Finally, we provide some computer simulations to show the capability of the scheme to conserve/dissipate the energy. Various fractional problems with functional forms of the anomalous diffusion of the solution are considered to that effect.
Highlights
The design of energy-preserving methods for physical systems has been a fruitful avenue of research in the last decades
Various seminal papers by Vázquez and his coworkers were published in the 1990s, including various energy-conserving numerical schemes to solve partial differential equations such as the Schrödinger equation [4], the sine-Gordon equation [5,6], the Klein–Gordon equations [7], and even systems consisting of ordinary differential equations [8]
The model under investigation includes the presence of constant damping, a nonlinear reaction term and general functions that depend on anomalous diffusion forms of the solution
Summary
The design of energy-preserving methods for physical systems has been a fruitful avenue of research in the last decades. Some energy-preserving methods have been proposed to simulate the nonlinear dynamics of three-dimensional beams undergoing finite rotations [9], to approximate the kinematics of geometrically exact rods [10] and to design algorithms for frictionless dynamic contact problems [11] After those seminal works by Vázquez and coauthors, the investigation on energy-preserving schemes became a vast area of research. They contributed to the state of the art by reviewing various existing methods for hyperbolic partial differential equations, which conserved or dissipated the energy of the systems [14,15] Those works would eventually pave the road to the birth of the discrete variational derivative method, which is a helpful tool to construct finite-difference schemes resembling he variational properties of continuous models [16].
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