An implicit four-step computational method in the study on the effects of damping in a modified α-Fermi–Pasta–Ulam medium

We present an implicit finite-difference scheme to approximate solutions of generalized α-Fermi–Pasta–Ulam systems defined on bounded domains which, amongst other features, include the presence of external and internal damping. Both continuous and semi-discrete media are considered in this paper, and several other scalar parameters are considered in the mathematical model. The numerical method is consistent with the problems under study, and it has a discrete energy scheme associated with it. It is shown that the method consistently approximates the continuous rate of change of energy of the mathematical problem with respect to time and, as a corollary, we obtain that the method is conservative when the damping coefficients are equal to zero, and the boundary points either are fixed or satisfy null Neumann conditions. We briefly state the computational details of the implementation, and simulations showing the validity of our method are provided in this work. As a result, we observe that our method preserves the energy of conservative systems at a high degree of accuracy. Finally, we present numerical experiments that evidence the effects of the presence of the damping coefficients in the problem that originated the investigation of α-Fermi–Pasta–Ulam chains more than 50 years ago.