Abstract
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 Cesàro mean. In this direction, we observe the numerical evidence of the almost convergence of the wave speeds for the Burridge–Knopoff model of earthquakes.
Highlights
Earthquakes occur along fractures in the Earth’s crust, named faults, characterized by a steady accumulation of tension, when big quantities of energy are suddenly released due to the relative motion between the two sides involved
We have performed numerical simulations aimed at providing a comprehensive overview of the typical dynamics the Burridge–Knopoff model is ruled by, through an effective numerical method based on a Predictor–Corrector strategy
Speaking about the traveling fronts theory, the numerical evidence of a limit for the wave speed averages paves the way to state the existence of propagating fronts starting from the specific discrete version of the Burridge–Knopoff model provided with the velocity-weakening friction
Summary
Earthquakes occur along fractures in the Earth’s crust, named faults, characterized by a steady accumulation of tension, when big quantities of energy are suddenly released due to the relative motion between the two sides involved. The existence of forces able to solicit plates is an important factor to explain seismic events, nothing would happen if friction did not inhibit the relative motion between the two different sides of an active fault Connected with these concepts is the stick–slip phenomenon, associated with the earthquakes by Brace and Byerlee [7]. The borders of a fault exhibit asperities which make the local slip very difficult: As a consequence, tension increases and the motion is inhibited by the balance between tension and friction Once this equilibrium is compromised, due to the steady accumulation of stress, a slip of the sides involved occurs and a great quantity of energy is released, generating an earthquake.
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