514 IGPM514.pdf        June 2020
TITLE Higher-dimensional 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 non-intrusive 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 L1-error of the stochastic moments by the L1-error of the approximation is proven. For the numerical approximation a Runge-Kutta discontinuous Galerkin method is employed and a multi-element 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