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12.03.2020 Open Access

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

BIT Numerical Mathematics
Jan Giesselmann, Fabian Meyer, Christian Rohde
Wichtige Hinweise
Communicated by Jan Nordström.
F.M., C.R. thank the Baden–Württemberg Stiftung for support via the Project ’SEAL’. J.G. thanks the German Research Foundation (DFG) for support of the Project via DFG Grant GI1131/1-1.

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This article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented. For the stochastic discretization, a non-intrusive approach, namely the Stochastic Collocation method is used. The spatio-temporal discretization relies on the Runge–Kutta Discontinuous Galerkin method. The a posteriori estimator is derived using smooth 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. The scaling properties of the residuals are investigated and the efficiency of the proposed adaptive algorithms is illustrated in various numerical examples.

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