# A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

## TL;DR

This paper develops an a posteriori error analysis and adaptive non-intrusive numerical schemes for one-dimensional random hyperbolic conservation laws, enabling efficient and reliable simulations with stochastic and spatial adaptivity.

## Contribution

It introduces a novel residual-based adaptive mesh refinement algorithm for stochastic conservation laws using the Stochastic Collocation method and Runge--Kutta Discontinuous Galerkin discretization.

## Key findings

- The error estimator effectively separates stochastic and deterministic errors.
- The adaptive algorithm improves computational efficiency in numerical experiments.
- Numerical results demonstrate the estimator's accuracy and the method's robustness.

## Abstract

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.