A computational technique with multiple properties of consistency in the study of modified β-Fermi-Pasta-Ulam chains

In this work, we present a numerical method to approximate solutions of a continuous β-Fermi-Pasta-Ulam-like medium which, amongst other features, includes the presence of damping and a fourth-order partial derivative, obtained when approximating the continuous limit of the classical β-Fermi-Pasta-Ulam chain. The method includes a discrete energy scheme which consistently approximates the continuous energy formula; moreover, it is shown that the discrete rate of change of energy consistently approximates its continuous counterpart. Computational experiments are conducted in order to show that the method preserves energy in the case of conservative systems, and it loses energy in the case of dissipative media. An application is conducted in order to approximate the occurrence of the process of nonlinear supratransmission in the classical β-Fermi-Pasta-Ulam chain and in discrete sine-Gordon systems, obtaining results which are in good agreement with the theory available in the literature.