A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
This work provides a theoretical foundation and practical adaptive algorithm for uncertainty quantification in hyperbolic conservation laws, benefiting computational scientists modeling random inputs in fluid dynamics and related fields.
The authors establish existence and uniqueness of random entropy admissible solutions for one-dimensional random systems of hyperbolic conservation laws, and derive an a posteriori error estimator for numerical approximations using Stochastic Collocation and Runge-Kutta Discontinuous Galerkin methods. The estimator enables a novel residual-based space-stochastic adaptive mesh refinement algorithm, demonstrated with numerical examples.
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.