  In this paper, we study the stochastic collocation (SC) methods for
uncertainty quantification (UQ) in hyperbolic systems of nonlinear partial
differential equations (PDEs). In these methods, the underlying PDEs are
numerically solved at a set of collocation points in random space. A standard
SC approach is based on a generalized polynomial chaos (gPC) expansion, which
relies on choosing the collocation points based on the prescribed probability
distribution and approximating the computed solution by a linear combination of
orthogonal polynomials in the random variable. We demonstrate that this
approach struggles to accurately capture discontinuous solutions, often leading
to oscillations (Gibbs phenomenon) that deviate significantly from the physical
solutions. We explore alternative SC methods, in which one can choose an
arbitrary set of collocation points and employ shape-preserving splines to
interpolate the solution in a random space. Our study demonstrates the
effectiveness of spline-based collocation in accurately capturing and assessing
uncertainties while suppressing oscillations. We illustrate the superiority of
the spline-based collocation on two numerical examples, including the inviscid
Burgers and shallow water equations.
