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Error control for statistical solutions of hyperbolic systems of conservation laws. (English) Zbl 1503.65227

The paper deals with hyperbolic systems of conservation laws and provides a reliable a posteriori error estimator, i.e., a computable upper bound for the numerical approximation error of dissipative statistical solutions is proposed in one spatial dimension. To this end, regularized empirical measures are used and the so-called relative entropy method by C. M. Dafermos and R. J. DiPerna [J. Differ. Equations 20, 90–114 (1976; Zbl 0323.35050)] can be applied. The error estimator is split into a stochastic and a spatiotemporal part. The Wasserstein distance between dissipative statistical solutions and the numerical approximation computed with a Runge-Kutta DG method in one spatial dimension is determined. Different numerical experiments illustrate the theoretical results, including a numerical approximation of the Wasserstein distance and an application to smooth and non-smooth solutions.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
62G30 Order statistics; empirical distribution functions
76N15 Gas dynamics (general theory)
35Q31 Euler equations

Citations:

Zbl 0323.35050

Software:

GitHub; POT
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Full Text: DOI

References:

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