Implicit Euler simulation of one-dimensional Burridge-Knopoff model of earthquakes with set-valued friction laws

In the simulations of Burridge-Knopoff (BK) model of earthquakes, the friction force laws are important to produce earthquake-like stick-slip behaviors. Some friction force laws are set-valued and the BK model with them can produce consistent results with observed data of earthquakes in some aspects. However, it is cumbersome to simulate the BK model with set-valued laws by conventional explicit integration methods. In the presence of set-valued laws, the explicit integration methods can easily lead to the numerical chattering, violated constraints on the velocity of force laws, and the difficulty of identifying the states of blocks of the BK model. This paper employs an implicit Euler integration method to simulate the BK model with symmetric and asymmetric set-valued laws. This method removes the numerical chattering in the BK model, even in the cases of large time step sizes. It can easily detect the stuck or slipping state of a block element. Comparing to previous results integrated by explicit integration methods in the literature, the results integrated by this implicit method show smoother curves and lower irregularities in the magnitude distribution of events.