Uncertainty quantification of multidimensional dynamical systems based on adaptive numerical solutions of the Liouville equation

Propagation of uncertainty in multidimensional dynamical systems, in the presence of parametric uncertainties, can be quantified by the solution of the underlying Liouville equation that governs the evolution of a multivariate joint probability density function of random variables associated with states and parameters. In this paper we propose an efficient numerical solution of the Liouville equation that involves (a) sampling at Gauss-quadrature nodes of random variables corresponding to uncertain parameters and (b) evolution of the associated conditional probability density functions using a finite difference method with time-adaptive computational domains in multiple dimensions. The proposed approach is designed to accurately predict long-time statistics of random variables corresponding to system states, including moments and probability density function, for dynamical systems of moderate dimension. The proposed approach is applied to four different dynamical systems, including (i) single spring-mass system, (ii) Van der Pol oscillator, (iii) double spring-mass system and (iv) a typical section nonlinear aeroelastic model. When compared to a conventional finite difference based numerical solution on a fixed grid, the solution obtained from the proposed adaptive grid based approach involves a considerable reduction in the required number of grid points for equivalent accuracy. For the single spring-mass system, for which an analytical solution is found, comparison with Monte Carlo simulation results indicates that the proposed adaptive numerical solution approach is generally one to two orders of magnitude more computationally efficient for a given level of accuracy.