Bound preserving {P}oint-{A}verage-{M}oment {P}olynomi{A}l-interpreted ({PAMPA}) on polygonal meshes

Abstract

We present a novel discretisation strategy, strongly inspired from Roe's Active Flux scheme. It can use polygonal meshes and is provably bound preserving for scalar problems and the Euler equations. Several cases demonstrates the quality of the method, and improvements with respect to previous work of the authors. This paper is a summary of \cite{BPPampa}.

Figures (6)

• Figure 1: Sub-triangulation that is used to define a low order scheme. The point values on the boundary of $P$ are numbered clockwise modulo $N_P$ where $N_P$ is the number of DoFs on the boundary. The point $\mathbf{y}_P$ is a point for which $P$ is star shaped, usually the centroid of $P$ in most applications. On the right panel are shown the normals.
• Figure 2: Definition of $e^\pm$ and $\mathbf{n}_\sigma^\pm$: (a) case of a vertex, (b) case of a non vertex. We use $\mathbf{n}_\sigma=\mathbf{n}_\sigma^++\mathbf{n}_\sigma^-$.
• Figure 3: Zalesak test case, after 1 rotation. 20 isolines of (a): the point values (b):the average, and (d): the exact solution. CFL=0.4, $4709$ nodes and $23704$ elements. On (c) is represented the mesh and the average isolines
• Figure 4: 30 equi-spaced isolines of the density as in Figure \ref{['KT_old']}, cfl=0.4 on a $100\times 100$ mesh. (a): averaged values, (b): point values.
• Figure 5: 30 equi-spaced isolines of the density as in Figure \ref{['KT_old']}, cfl=0.4 on a $400\times 400$ mesh. (a): averaged values, (b): point values.

Theorems & Definitions (1)

• Remark 6.1