A Numerical Scheme for Various Nonlinear Forces, Including Collisions, Which Does Not Require an Iterative Root Finder
Nonlinear forces are ubiquitous in physical systems, and of prominent importance in musical acoustics. Though many models exist to describe such forces, in most cases the associated numerical schemes rely on iterative root finding methods, such as NewtonRaphson or gradient descent, which are computationally expensive and which therefore could represent a computational bottleneck. In this paper, a model for a large class of nonlinear forces is presented, and a novel family of energy-conserving finite difference schemes given. The schemes only require the evaluation of the roots of a quadratic function. A few applications in the lumped case are shown, and the robustness and accuracy of the scheme tested.