Numerical Methods for a Nonlinear Impact Model: A Comparative Study With Closed-Form Corrections

A physically based impact model-already known and exploited in the field of sound synthesis-is studied using both analytical tools and numerical simulations. It is shown that the Hamiltonian of a physical system composed of a mass impacting on a wall can be expressed analytically as a function of the mass velocity during contact. Moreover, an efficient and accurate approximation for the mass outbound velocity is presented, which allows to estimate the Hamiltonian at the end of the contact. Analytical results are then compared to numerical simulations obtained by discretizing the system with several numerical methods. It is shown that, for some regions of the parameter space, the trajectories of the discretized systems may significantly drift from the analytically derived curves. Two approaches, based on enforcing numerical energy consistency, are then proposed to improve the accuracy of numerical simulations.