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IGPM514.pdf June 2021 
TITLE 
Higherdimensional deterministic formulation of hyperbolic conservation laws with uncertain initial data 
AUTHORS 
Michael Herty, Adrian Kolb, Siegfried Müller 
ABSTRACT 
We discuss random hyperbolic conservation laws and introduce a novel formulation interpreting
the stochastic variables as additional spatial dimensions with zero flux. The approach is
compared with established nonintrusive approaches to random conservation laws. In the scalar case,
an entropy solution is proven to exist if and only if a random entropy solution for the original problem
exists. Furthermore, existence and numerical convergence of stochastic moments is established. Along
with this, the boundedness of the L1error of the stochastic moments by the L1error of the approximation is proven. For the numerical approximation a RungeKutta discontinuous Galerkin method is employed and a multielement stochastic collocation is used for the approximation of the stochastic moments. By means of grid adaptation the computational effort is reduced in the spatial as well as in the stochastic directions, simultaneously. Results on Burger's and Euler equation are validated by several numerical examples and compared to Monte Carlo simulations. 
KEYWORDS 
hyperbolic conservation laws, uncertainty quantification, discontinuous Galerkin methods, stochastic collocation, multiresolution analysis 
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