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

We develop a general polynomial chaos (gPC) based stochastic Galerkin (SG) for hyperbolic equations with random and singular coefficients. Due to the singular nature of the solution, the standard gPC-SG methods may suffer from a poor or even non convergence. Taking advantage of the fact that the discrete solution, by the central type finite difference or finite volume approximations in space and time for example, is smoother, we first discretize the equation by a smooth finite difference or finite volume scheme, and then use the gPC-SG approximation to the discrete system. The jump condition at the interface is treated using the immersed upwind methods introduced in Jin (Proc Symp Appl Math 67(1):93-104, 2009) and Jin and Wen (Commun Math Sci 3:285-315, 2005). This yields a method that converges with the spectral accuracy for finite mesh size and time step. We use a linear hyperbolic equation with discontinuous and random coefficient, and the Liouville equation with discontinuous and random potential, to illustrate our idea, with both one and second order spatial discretizations. Spectral convergence is established for the first equation, and numerical examples for both equations show the desired accuracy of the method.