Entropy stable shock capturing for high-order DGSEM on moving meshes

TL;DR

The paper tackles stable, high-order simulation of compressible flows on time-dependent domains by developing a hybrid entropy-stable DGSEM with a convex blending to a low-order FV subcell scheme within an ALE framework. It employs Gauss--Lobatto nodes and a diagonal-norm SBP structure to enable flux differencing and entropy considerations, while enforcing the discrete geometric conservation law on moving meshes. Numerical tests demonstrate free-stream preservation, expected h- and p-convergence on moving grids, and entropy stability alongside effective shock capturing, including a piston-tube scenario. The approach integrates SBP properties, entropy-conservative fluxes, and subcell blending to deliver robust, high-order accurate simulations on moving domains, with an open-source implementation in FLEXIK.

Abstract

In this paper, a shock capturing for high-order entropy stable discontinuous Galerkin spectral element methods on moving meshes is proposed using Gauss--Lobatto nodes. The shock capturing is achieved via the convex blending of the high-order scheme with a low-order finite volume subcell operator. The free-stream and convergence properties of the hybrid scheme are demonstrated numerically along with the entropy stability and shock capturing capabilities.

Figures (2)

• Figure 1: Validation of the spatial discretization with curved faces for a static (DG ($\nu_{\@nil} =0$)) and sinusoidally deformed domain (DG). Left: $p$-convergence on a $4^3$ grid with $\mathcal{N} \in [1,10]$. Right: $h$-convergence for $\mathcal{N}=4$. Grid sequence ranges from $2^3$ up to $32^3$.
• Figure 2: Left: Temporal evolution of the integral entropy conservation errors for the Euler equations using the TGV as a test case with and without additional surface dissipation. Right: Density distribution at $t=11$ for the moving piston test case. The dashed lines highlight the position of the piston.