Uncertainty Quantification for Systems with Random Initial Conditions Using Wiener–Hermite Expansions

A number of engineering problems, including laminar‐turbulent transition in convectively unstable flows, require predicting the evolution of a nonlinear dynamical system under uncertain initial conditions. The method of Wiener–Hermite expansion is an attractive alternative to modeling methods, which solve for the joint probability density function of the stochastic amplitudes. These problems include the “curse of dimensionality” and closure problems. In this paper, we apply truncated Wiener–Hermite expansions with both fixed and time‐varying bases to a model stochastic system with three degrees of freedom. The model problem represents the combined effects of quadratic nonlinearity and stochastic initial conditions in a generic setting and occurs in related forms in both classical dynamics, turbulence theory, and the nonlinear theory of hydrodynamic stability. In this problem, the truncated Wiener–Hermite expansions give a good account of short‐time behavior, but not of the long‐time relaxation characteristic of this system. It is concluded that successful application of truncated Wiener–Hermite expansions may require special adaptations for each physical problem.