5 research outputs found
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
In this article we consider one-dimensional random systems of hyperbolic
conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws
which involve random initial data and random flux functions. Based on these
results we present an a posteriori error analysis for a numerical approximation
of the random entropy admissible solution. For the stochastic discretization,
we consider a non-intrusive approach, the Stochastic Collocation method. The
spatio-temporal discretization relies on the Runge--Kutta Discontinuous
Galerkin method. We derive the a posteriori estimator using continuous
reconstructions of the discrete solution. Combined with the relative entropy
stability framework this yields computable error bounds for the entire
space-stochastic discretization error. The estimator admits a splitting into a
stochastic and a deterministic (space-time) part, allowing for a novel
residual-based space-stochastic adaptive mesh refinement algorithm. We conclude
with various numerical examples investigating the scaling properties of the
residuals and illustrating the efficiency of the proposed adaptive algorithm
Error control for statistical solutions
Statistical solutions have recently been introduced as a an alternative
solution framework for hyperbolic systems of conservation laws. In this work we
derive a novel a posteriori error estimate in the Wasserstein distance between
dissipative statistical solutions and numerical approximations, which rely on
so-called regularized empirical measures. The error estimator can be split into
deterministic parts which correspond to spatio-temporal approximation errors
and a stochastic part which reflects the stochastic error. We provide numerical
experiments which examine the scaling properties of the residuals and verify
their splitting.Comment: 25 pages, 2 figure