Analysis of the leapfrog-Verlet method applied to the Kuwabara-Kono force model in discrete element method simulations of granular materials

The discrete element method (DEM) is a numerical technique widely used to simulate granular materials. The temporal evolution of these simulations is often performed using a Verlet-type algorithm, because of its second order and its desirable property of better energy conservation. However, when dissipative forces are considered in the model, such as the nonlinear Kuwabara-Kono model, the Verlet method no longer behaves as a second order method, but instead its order decreases to 1.5. This is caused by the singular behavior of the derivative of the damping force in the Kuwabara-Kono model at the beginning of particle collisions. In this work, we introduce a simplified problem which reproduces the singularity of the Kuwabara-Kono model and prove that the order of the method decreases from 2 to \(1+q\), where \(0< q < 1\) is the exponent of the nonlinear singular term.