A superconvergent convolution quadrature method for time-dependent dynamical energy analysis

Dynamical Energy Analysis (DEA) was introduced in 2009 as a novel method for predicting high-frequency acoustic and vibrational energy distributions [1]. In this work we detail how DEA can be reformulated in the time-domain by means of a convolution integral operator and apply a modified multistep-based Convolution Quadrature (CQ) method to discretise in time. Stable time-stepping based on multistep CQ schemes has typically been limited to second-order convergence as a consequence of the second Dahlquist barrier. In this work we apply a simple modification to the traditional second-order backward-difference multistep scheme, which leads to third-order convergence of the time discretisation in time-dependent DEA. The final result is a fully time-dependent DEA method that can accurately track the propagation of high-frequency transient signals through phase-space.