A Well-Balanced Stochastic Galerkin Method for Scalar Hyperbolic Balance Laws with Random Inputs

We propose a generalized polynomial chaos (gPC) based stochastic Galerkin methods for scalar hyperbolic balance laws with random geometric source terms or random initial data. This method is well-balanced (WB), in the sense that it captures the stochastic steady state solution with high order accuracy. The framework of the stochastic WB schemes is presented in details, along with several numerical examples to illustrate their accuracy and effectiveness. The goal of this paper is to show that the stochastic WB scheme yields a more accurate numerical solution at steady state than the non-WB ones.