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 artefacts arising from anomalous auxiliary variable behaviour in the simulation of a specific nonlinear transverse string vibration model. This work revisits that model using SAV for time integration, but with modal discretisation in space. This approach benefits from reduced numerical dispersion and facilitates the implementation of refined loss profiles. A convergence analysis is presented against a reference solution, demonstrating the accuracy of the modal solver and verifying its behaviour under reduced dispersion conditions. Additionally, a constraint technique adapted from collision modelling is applied to the auxiliary variable, significantly improving simulation performance at audio sample rate.