Implicit and explicit schemes for energy-stable simulation of string vibrations with collisions: Refinement, analysis, and comparison

Abstract

Collisions play an important role in the sound production mechanism of musical string instruments. Impactive contact modelling must therefore be incorporated in energy-stable time-stepping schemes for simulation and sound synthesis. Recently, explicit and linearly implicit schemes based on energy quadratisation have emerged as a means of numerical modelling of nonlinear string vibrations that avoids the need for iterative solvers, which significantly enhances the applicability to real-time sound synthesis. However when applied to modelling collisions, spurious oscillations and artefacts have been reported, which can be linked to the one-sided nature of the contact force. To investigate whether and how these limitations can be avoided, this paper proposes refined numerical scheme variants for both lumped and distributed contact configurations, and assesses their accuracy and efficiency. For reference, two nonlinearly implicit schemes are also included in the comparisons. Guaranteed correctness of the sign of the numerical contact force is proven for several of the scheme variants. Numerical experiments across three case studies indicate that each of the proposed schemes converges towards the exact solution, and that for sufficiently small time steps, explicit schemes generally perform well compared to implicit schemes.