Stabilizing DG Methods Using Dafermos' Entropy Rate Criterion: III -- Unstructured Grids

Abstract

The approach presented in the second installment of this series is extended to multidimensional systems of conservation laws that are approximated via a Discontinuous Galerkin method on unstructured (triangular) grids. Special attention is paid to predicting the entropy dissipation from boundaries. The resulting schemes are free of tunable viscosity parameters and tested on the Euler equations. The trinity of testcases is the spreading of thermal energy from a point source, transsonic and supersonic flows around airfoils, and supersonic air inlets.

Figures (16)

• Figure 1: Selected nodes for polynomial degrees $1$ and $3$ and the map from the reference element to the physical element.
• Figure 2: Assumptions on weak solutions. One can construct weak solutions that violate these assumptions, but these are not physically relevant solutions from our point of view.
• Figure 3: Subdivision of a triangle into finer triangles
• Figure 4: A single step of a scheme formulated on a grid of cells $Z_i \in \mathcal{Z}$ can be rephrased into a set of decoupled schemes for a set of quadrilaterals $Q_{ji}$. After the centroid of a cell $Z_i$ was joined with its vertices results a subdivision of the cell into triangular subcells $Z_{ij}$ that are only adjacent to the subcell $Z_{ji}$ of another cell $Z_j$ and other subcells of $Z_i$. We can assign to every subcell a quadrilateral cell $Q_{ij}$ of equivalent size. The fluxes over the internal, dotted lines, can be set as boundary conditions on the dotted boundaries of the quadrilaterals.
• Figure 5: The two natural boundary conditions for hyperbolic conservation laws that are modelling fluid flows.

Theorems & Definitions (4)

• Definition 1: Discrete per cell entropy
• Conjecture 1
• Definition 2: Positive, conservative filter
• Definition 3: Positive, conservative, discrete filter