Adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws

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 a Roe-type solver. The adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain. In addition, the stochastic features of the solution greatly vary in space and time so that the constructed stochastic approximation space depends on space and time. The dynamically evolving stochastic discretization uses a tree-structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic multiresolution analysis. Efficiency of the overall adaptive scheme is assessed on a stochastic nonlinear conservation law with uncertain initial conditions and velocity leading to expansion waves and shocks that propagate with random velocities. Numerical tests highlight the computational savings achieved as well as the benefit of using anisotropic discretizations in the context of problems involving a large number of stochastic parameters.