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Numerical solution of scalar conservation laws with random flux functions
(2012)Research ReportWe consider scalar hyperbolic conservation laws in several space dimensions, with a class of random (and parametric) flux functions. We propose a Karhunen–Loève expansion on the state space of the random flux. For random flux functions which are Lipschitz continuous with respect to the state variable, we prove the existence of a unique random entropy solution. Using a Karhunen–Loève spectral decomposition of the random flux into principal ...Report 
Multilevel Monte Carlo finite difference and finite volume methods for stochastic linear hyperbolic systems
(2012)Research ReportWe consider stochastic multidimensional linear hyperbolic systems of conservation laws. We prove existence and uniqueness of a random weak solution, provide estimates for the regularity of the solution in terms of regularities of input data, and show existence of statistical moments. Bounds for mean square error vs. expected work are proved for the MultiLevel Monte Carlo Finite Volume algorithm which is used to approximate the moments ...Report 
Multilevel Monte Carlo finite difference and finite volume methods for stochastic linear hyperbolic systems
(2012)We consider stochastic multidimensional linear hyperbolic systems of conservation laws. We prove existence and uniqueness of a random weak solution, provide estimates for the regularity of the solution in terms of regularities of input data, and show existence of statistical moments. Bounds for mean square error vs. expected work are proved for the MultiLevel Monte Carlo Finite Volume algorithm which is used to approximate the moments ...Report 
Multilevel Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws
(2012)Research ReportReport 
Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes
(2012)Research ReportFlow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pressure exists and is a weak solution of the ...Report