Abstract
In this chapter, we introduce spatial discretization schemes for systems of conservation laws. For smooth problems, summation-by-parts operators with weak enforcement of boundary conditions allow for the design of stable high-order accurate schemes. Summation by parts is the discrete equivalent of integration by parts and the matrix operators that are presented lead to energy estimates that in turn lead to provable stability. The discrete stability analysis follows naturally from the continuous analysis of well-posedness. For non-smooth problems, the need to accurately capture multiple solution discontinuities of hyperbolic stochastic Galerkin systems requires the introduction of shock-capturing methods. In this setting, we outline the MUSCL scheme with flux limiters and the HLL Riemann solver. We also briefly discuss how to add artificial dissipation and an issue regarding time integration.
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Pettersson, M.P., Iaccarino, G., Nordström, J. (2015). Numerical Solution of Hyperbolic Problems. In: Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-10714-1_4
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DOI: https://doi.org/10.1007/978-3-319-10714-1_4
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