A local time--dependent Generalized Polynomial Chaos method for Stochastic Dynamical Systems

Abstract

Generalized Polynomial Chaos (gPC) is known to fail for problems involving strong nonlinear dependencies on stochastic inputs, which especially arise in the context of long term integration or stochastic discontinuities. There are various attempts in the literature which address these difficulties, such as the time-dependent generalized Polynomial Chaos (TD-gPC) and the multi-element generalized Polynomial Chaos (ME-gPC) both leading to higher accuracies but higher numerical costs in comparison to the classical gPC approach. A combination of these methods is introduced, which leads to a powerful solution method since high accuracies can be maintained and computational cost can be distributed by utilizing parallel computation. However, to be able to apply the hybrid method to all types of ordinary differential equations subject to random inputs, new modifications with respect to TD-gPC are carried out by creating an orthogonal tensor basis consisting of the random input variable as well as the solution itself. Such modifications allow TD-gPC to capture the dynamics of the solution by increasing the approximation quality of its time derivatives.