Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations

Abstract

We consider time-invariant linear systems of differential algebraic equations, which include physical parameters or other parameters. Uncertainties of the parameters are modelled by random variables. We expand the corresponding random-dependent solutions in the polynomial chaos. Approximations of unknown coefficient functions can be obtained by quadrature or sampling schemes. Alternatively, stochastic collocation methods or the stochastic Galerkin approach yield larger coupled systems of differential algebraic equations. We show the equivalence of these types of numerical methods under certain assumptions. The index of the coupled systems is analysed in comparison to the original systems. Sufficient conditions for an identical index are derived. Furthermore, we present results of numerical simulations for an example.