Convergence analysis and relaxation techniques for modal scalar auxiliary variable methods applied to nonlinear transverse string vibration

Many models of string vibration are available in the literature, playing a fundamental role in musical acoustics and sound synthesis. In particular, models incorporating geometric nonlinearities due to large strains enable the reproduction of important perceptual features. For real-time synthesis, ensuring the stability and efficiency of the numerical algorithms is crucial. To this end, explicit schemes based on the scalar auxiliary variable (SAV) method have been proposed, offering the advantage of avoiding iterative solvers by introducing an auxiliary variable into the system. However, a recent study identified spurious artifacts arising from anomalous auxiliary variable behavior in the simulation of a specific nonlinear transverse string vibration model. This work revisits the same model. Discretization in space is performed using finite difference and modal methods, the latter benefitting from reduced numerical dispersion when the linear part is integrated in time using exact numerical techniques. The nonlinear part is integrated using SAV. A convergence analysis is conducted against a reference solution to check the accuracy properties of both approaches. Additionally, a novel technique adapted from collision modeling is proposed to constrain the auxiliary variable, significantly improving simulation performance at lower sample rates.