Mimetic Metrics for the DGSEM

TL;DR

The paper tackles free-stream preservation and entropy stability for DGSEM on curved grids by ensuring the discrete metric terms are divergence-free using a mimetic, FEEC-based approach. It constructs metric terms via a de Rham cochain framework and two projection routes, guaranteeing $div(p^3(...))=0$ and thus stability. Numerical tests on the 3D compressible Euler equations with curved geometry show that the mimetic method preserves free-streams with smaller rounding errors (notably in $L_ obreaksp extsuperscript{∞}$) and metric-term errors converge to machine precision, while in 2D it aligns with Kopriva's curl form. The approach generalizes to other element types (triangles, tetrahedra) via FEEC, enhancing robustness of DG simulations on curved domains.

Abstract

Free-stream preservation is an essential property for numerical solvers on curvilinear grids. Key to this property is that the metric terms of the curvilinear mapping satisfy discrete metric identities, i.e., have zero divergence. Divergence-free metric terms are furthermore essential for entropy stability on curvilinear grids. We present a new way to compute the metric terms for discontinuous Galerkin spectral element methods (DGSEMs) that guarantees they are divergence-free. Our proposed mimetic approach uses projections that fit within the de Rham Cohomology.

Figures (5)

• Figure 1: Diagram of the continuous and discrete 3D de Rham complex
• Figure 2: Diagram of the continuous and discrete 1D de Rham complex
• Figure 3: Diagram of the continuous and discrete 2D de Rham complex
• Figure 4: Free-stream preservation errors of $\rho e$ with method of Kopriva kopriva, Eq. \ref{['eq:3dmetricdiscrete']}, and the Mimetic approach, Eq. \ref{['eq:op_blue']}.
• Figure 5: Error in the metric terms with method of Kopriva, \ref{['eq:3dmetricdiscrete']}, and the Mimetic approach, see \ref{['eq:op_blue']}.

Theorems & Definitions (2)

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