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In this paper, we develop new high-order numerical methods for hyperbolicsystems of nonlinear partial differential equations (PDEs) with uncertainties.The new approach is realized in the semi-discrete finite-volume framework andit is based on fifth-order weighted essentially non-oscillatory (WENO)interpolations in (multidimensional) random space combined with second-orderpiecewise linear reconstruction in physical space. Compared with spectralapproximations in the random space, the presented methods are essentiallynon-oscillatory as they do not suffer from the Gibbs phenomenon while stillachieving a high-order accuracy. The new methods are tested on a number ofnumerical examples for both the Euler equations of gas dynamics and theSaint-Venant system of shallow-water equations. In the latter case, the methodsare also proven to be well-balanced and positivity-preserving.

New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties