Numerical evidences of almost convergence of wave speeds for the Burridge-Knopoff model

This paper deals with the numerical approximation of a stick-slip system, known in the literature as Burridge-Knopoff model, proposed as a simplified description of the mechanisms generating earthquakes. Modelling of friction is crucial and we consider here the so-called velocity-weakening form. The aim of the article is twofold. Firstly, we establish the effectiveness of the classical Predictor-Corrector strategy. To our knowledge, such approach has never been applied to the model under investigation. In the first part, we determine the reliability of the proposed strategy by comparing the results with a collection of significant computational tests, starting from the simplest configuration to the more complicated (and more realistic) ones, with the numerical outputs obtained by different algorithms. Particular emphasis is laid on the Gutenberg-Richter statistical law, a classical empirical benchmark for seismic events. The second part is inspired by the result by Muratov (Phys Rev 59:3847-3857, 1999) providing evidence for the existence of traveling solutions for a corresponding continuum version of the Burridge-Knopoff model. In this direction, we aim to find some appropriate estimate for the crucial object describing the wave, namely its propagation speed. To this aim, motivated by LeVeque and Yee (J Comput Phys 86:187-210, 1990) (a paper dealing with the different topic of conservation laws), we apply a space-averaged quantity (which depends on time) for determining asymptotically an explicit numerical estimate for the velocity, which we decide to name LeVeque-Yee formula after the authors' name of the original paper. As expected, for the Burridge-Knopoff, due to its inherent discontinuity of the process, it is not possible to attach to a single seismic event any specific propagation speed. More regularity is expected by performing some temporal averaging in the spirit of the Cesaro mean. In this direction, we observe the numerical evidence of the almost convergence of the wave speeds for the Burridge-Knopoff model of earthquakes.