Simulation of the geometrically exact nonlinear string via energy quadratisation

String vibration represents an active field of research in acoustics. Small-amplitude vibration is often assumed, leading to simplified physical models that can be simulated efficiently. However, the inclusion of nonlinear phenomena due to larger string stretchings is necessary to capture important features, and efficient numerical algorithms are currently lacking in this context. Of the available techniques, many lead to schemes which may only be solved iteratively, resulting in high computational cost, and the additional concerns of existence and uniqueness of solutions. Slow and fast waves are present concurrently in the transverse and longitudinal directions of motion, adding further complications concerning numerical dispersion. This work presents a linearly-implicit scheme for the simulation of the geometrically exact nonlinear string model. The scheme conserves a numerical energy, expressed as the sum of quadratic terms only, and including an auxiliary state variable yielding the nonlinear effects. This scheme allows to treat the transverse and longitudinal waves separately, using a mixed finite difference/modal scheme for the two directions of motion. A matrix decomposition algorithm is presented, so to treat the sparse and full parts of the update matrix separately. Numerical experiments are presented throughout.