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We develop a general polynomial chaos (gPC) based stochastic Galerkin (SG)for hyperbolic equations with random and singular coefficients. Due to thesingu- lar nature of the solution, the standard gPC-SG methods may suffer froma poor or even non convergence. Taking advantage of the fact that the discretesolution, by the central type finite difference or finite volume approximationsin space and time for example, is smoother, we first discretize the equation bya smooth finite difference or finite volume scheme, and then use the gPC-SGapproximation to the discrete system. The jump condition at the interface istreated using the immersed upwind methods introduced in [8, 12]. This yields amethod that converges with the spectral accuracy for finite mesh size and timestep. We use a linear hyperbolic equation with discontinuous and randomcoefficient, and the Liouville equation with discontinuous and randompotential, to illustrate our idea, with both one and second order spatialdiscretizations. Spectral convergence is established for the first equation,and numerical examples for both equations show the desired accu- racy of themethod.

The Discrete Stochastic Galerkin Method for Hyperbolic Equations with Non-smooth and Random Coefficients