This paper deals with the design of adaptive anisotropic discretization schemes for conservation laws with stochastic parameters. A Finite Volume scheme is used for the deterministic discretization, while a piecewise polynomial representation is used at the stochastic level. The methodology is designed in the context of intrusive Galerkin projection methods with Roe-type solver. The adaptation aims at selecting the stochastic resolution level with regard to the local smoothness of the solution in the stochastic domain. In addition, the stochastic features of the solution greatly vary in the space and time so that the constructed stochastic approximation space depends on space and time. The overall method is assessed on the stochastic Burgers equation with shocks, showing significant computational savings.

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