An operator splitting based stochastic Galerkin method for the one-dimensional compressible Euler equations with uncertainty

We introduce an operator splitting based stochastic Galerkin method for the one-dimensional compressible Euler equations with random inputs. The method uses a generalized polynomial chaos approximation in the stochastic Galerkin framework (referred to as the gPC-SG method). It is well-known that such approximations for nonlinear system of hyperbolic conservation laws do not necessarily yield globally hyperbolic systems: the Jacobian may contain complex eigenvalues and thus trigger instabilities and ill-posedness. In this paper, we propose to split the underlying system into a linear hyperbolic system and two effectively scalar linear or nonlinear hyperbolic equations with variable coefficients and source terms. The gPC-SG method, when applied to each of these subsystems, results in globally hyperbolic systems. The performance of the new gPC-SG method is illustrated with a number of numerical examples with uncertainties from the initial data or equation of state.