A novel multilevel approach for the efficient computation of random hyperbolic conservation laws

A novel strategy for the efficient computation of moments to solutions of random hyperbolic conservation laws is presented. The random variables are considered as parameters to the equation but understood as additional (stochastic) directions. The key idea is to perform grid adaptation alternatingly in spatial and stochastic directions to compute the moments in stochastic direction and to select deterministically realizations, respectively. New realizations are computed on the adaptive spatial grid. For grid adaptation a multiresolution analysis is employed to perform data compression using different threshold values in spatial and stochastic directions. For proof of concept, the efficiency of the new approach is verified by the 1D–Euler equations with uncertain initial data. The results are compared with the classical Monte–Carlo method showing the improved performance of the proposed method.