An High-Order Entropy-Stable Shock Capturing on Heterogeneous Grids using Discontinuous Galerkin Spectral Element Methods
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The discontinuous Galerkin method allows a straightforward construction of efficient high-order schemes on arbitrary grids for hyperbolic partial differential equations (PDEs) such as the Euler equations. However, hyperbolic PDEs admit discontinuities in the solution, even if the initial solution is smooth. Moreover, it is well known that high-order operators induce oscillations at discontinuities, called Gibbs phenomena. Hence, adequate numerical treatment is necessary to detect and handle discontinuities. One example is a switching procedure where a troubled DG cell is switched to a low-order finite volume scheme, where the latter is defined on an equidistant point set. This approach enables a highly accurate and localized shock capturing. However, implementations of h-adaptive FV subcell schemes for DGSEM and related DG schemes were so far limited to hexahedral and tetrahedral elements. In this talk, a finite volume subcell shock capturing procedure for the entropy-stable discontinuous Galerkin spectral element method on heterogeneous grids, building on the work presented in [Keim2025], is proposed. The proposed scheme is based on an h-adaptive entropy-stable second-order total variational diminishing finite volume operator defined on an equidistant point set. The capabilities of the shock capturing procedure are demonstrated on a series of well-known numerical experiments including an application-oriented numerical problem.
