Numerical convergence of the scalar auxiliary variable method applied to nonlinear stiff string models.

Abstract

This work is concerned with the Scalar Auxiliary Variable (SAV) method applied to geometrically nonlinear string models, focusing on the analysis of numerical convergence across various model formulations. An ODE system with a potential akin to that of the geometrically exact string is first analysed, providing both mathematical and numerical insight. Then, three nonlinear stiff string models are considered: geometrically exact (in two formulations), cubic, and Kirchhoff-Carrier models, each incorporating a stiffness term based on the Euler-Bernoulli formulation. A unified mathematical description is introduced in the continuous domain, using both quadratically split and non-split forms. Spatial and temporal discretisation is performed using Finite-Difference Time-Domain (FDTD) methods. Convergence is assessed by comparing SAV solutions with benchmark results obtained from convergent reference integrators at high sample rates, considering the transverse displacement, longitudinal displacement (when present), and the auxiliary variable. Results show that the non-split SAV converges for all three models, but only when the spatial step remains above the stability threshold of the corresponding split version, despite the scheme being unconditionally stable; furthermore, the global error is further strongly dependent on a potential shift constant. The split formulation converges reliably only when the potential decomposition reflects a natural separation between linear and nonlinear dynamics, as in the cubic and Kirchhoff-Carrier models, and in one particular form of the geometrically exact string.