  This paper is concerned with generalized polynomial chaos (gPC) approximation
for a general system of quasilinear hyperbolic conservation laws with
uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized
with the aid of left eigenvector matrix of the Jacobian matrix. Stochastic
Galerkin method is then applied to derive the equations for the gPC expansion
coefficients. The resulting deterministic gPC Galerkin system is proved to be
symmetrically hyperbolic. This important property then allows one to use a
variety of numerical schemes for spatial and temporal discretization. Here a
higher-order and path-conservative finite volume WENO scheme is adopted in
space, along with a third-order total variation diminishing Runge-Kutta method
in time. The method is further extended to two-dimensional (2D) quasilinear
hyperbolic system with uncertainty, where the symmetric hyperbolicity of the
one-dimensional system is carried over via the operator splitting technique.
Several 1D and 2D numerical experiments are conducted to demonstrate the
accuracy and effectiveness of the proposed gPC stochastic Galerkin method.
