A family of lower- and higher-order transversal linearization techniques in non-linear stochastic engineering dynamics

Sample pathwise numerical integration of noise-driven engineering dynamical systems cannot generally be performed beyond a limited level of accuracy, especially when the noise processes are modelled using (filtered) white noises. Recently, a locally transversal linearization (LTL) strategy has been proposed by the author (Proc Roy Soc London A 2001; 457:539–566) for direct integration of deterministic and stochastic non-linear dynamical systems. The present effort is focussed on a host of extensions along with detailed theoretical error analyses of the linearization approach, especially as applied to problems in non-linear stochastic engineering dynamics. Thus, to begin with, estimates of local and global error orders in the basic LTL scheme are obtained separately for the displacement and velocity vectors when the system is driven either by a set of additive noises or by an arbitrary combination of (independently evolving) additive and multiplicative noises. Following this, a new family of higher-order LTL schemes is proposed in order to improve upon the basic LTL method and the associated error orders are established. A stepwise implementation of the lower- and higher-order versions of the LTL method, along with certain computational aspects, is also outlined. The proposed schemes are numerically illustrated, to a limited extent, for a single degree-of-freedom (SDOF) and a two degree-of-freedom (TDOF) non-linear engineering systems under additive and/or multiplicative white noise excitations