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.

离散元法 (DEM) 是一种广泛用于模拟颗粒材料的数值技术。这些模拟的时间演化通常使用 Verlet 型算法来执行，因为它是二阶算法并且具有更好的能量守恒的理想特性。然而，当模型中考虑耗散力时，例如非线性 Kuwabara-Kono 模型，Verlet 方法不再表现为二阶方法，而是其阶数降低至 1.5。这是由 Kuwabara-Kono 模型中粒子碰撞开始时阻尼力导数的奇异行为引起的。在这项工作中，我们引入了一个简化的问题，它再现了 Kuwabara-Kono 模型的奇点，并证明该方法的阶数从 2 减少到 \(1+q\)，其中 \(0< q < 1\) 是非线性奇异项的指数。