A semi-discrete Active Flux method for the Euler equations on Cartesian grids

Abstract

Active Flux is an extension of the Finite Volume method and additionally incorporates point values located at cell boundaries. This gives rise to a globally continuous approximation of the solution. Originally, the Active Flux method emerged as a fully discrete method, and required an exact or approximate evolution operator for the point value update. For nonlinear problems such an operator is often difficult to obtain, in particular for multiple spatial dimensions. We demonstrate that a new semi-discrete Active Flux method (first described in Abgrall&Barsukow, 2023 for one space dimension) can be used to solve nonlinear hyperbolic systems in multiple dimensions without requiring evolution operators. We focus here on the compressible Euler equations of inviscid hydrodynamics and third-order accuracy. We introduce a multi-dimensional limiting strategy and demonstrate the performance of the new method on both Riemann problems and subsonic flows.

Figures (22)

• Figure 1: Degrees of freedom employed in the third-order accurate Active Flux method on Cartesian grids. Stars denote point values, the cross denotes the cell average. Every cell has access to 8 point values and one average. Point values are shared with adjacent cells; per cell there is one average and $4 \cdot \frac{1}{4} + 4 \cdot \frac{1}{2} = 3$ point values.
• Figure 2: Left: An example of a plateau reconstruction. Here, $q_\text{NW} = 1$, $q_\text{W} = 1.35$, $q_\text{SW} = 0.6$, $q_\text{S} = 0.4$, $q_\text{SE} = 0$, $q_\text{E} = -0.2$, $q_\text{NE} = 0.0$, $q_\text{N} = 1$, $\bar{q} = 0.9$ (the S-edge is on the left and $\Delta x = \Delta y = 1$). All edges but the S-edge are reconstructed as hats, the S-edge is reconstructed parabolically. Right: A piecewise-biparabolic reconstruction of the same data; one clearly observes an overshoot. The isolines have a spacing of 0.1.
• Figure 3: Left: An example of a plateau reconstruction. $q_\text{NE} = 1, q_\text{NW} = 2, q_\text{SW} = -4, q_\text{SE} = 0, q_\text{N} = -1, q_\text{S} = 4, q_\text{W} = -5, q_\text{E} = -3, \bar{q} = 2$ (the W-edge is on the left and $\Delta x = \Delta y = 1$). All edges are reconstructed as hats. Right: A piecewise-biparabolic reconstruction of the same data; one clearly observes an overshoot. The isolines have a spacing of 0.25.
• Figure 4: Convergence study. Top left: Setup \ref{['eq:convergence1u']}--\ref{['eq:convergence1p']} at initial time and at $t=0.05$, shown as a scatter plot as a function of radius, computed on a $256\times 256$ grid. Bottom left: Setup \ref{['eq:convergence2u']}--\ref{['eq:convergence2p']} at initial time on a grid of $100 \times 100$, shown as a scatter plot. Right top: $\ell^1$ error of the numerical solution of the point values, i.e. the average of $\frac{1}{|\Omega|} \sum_{ij} |q_{i+\frac{1}{2},j+\frac{1}{2}}(t^n) - q_\text{ref}(t^n, x_{i+\frac{1}{2}}, y_{j+\frac{1}{2}})| \Delta x \Delta y$ for the setup \ref{['eq:convergence1u']}--\ref{['eq:convergence1p']}. The analogous error of the averages has virtually the same values and is not shown. Right bottom: Same for the setup \ref{['eq:convergence2u']}--\ref{['eq:convergence2p']}.
• Figure 5: Radial scatter plot of the two-dimensional version of Sod's shock tube solved on a $100 \times 100$ grid. The solid line shows a finely resolved solution of the one-dimensional, radial Euler equations obtained with a standard Finite Volume method. We show the pressure offset by 0.1 for better readability of the plot. Left: No limiting. Right: Limiting used. The limiting is successful at suppressing oscillations at the shock and around the rarefaction. As in a radial scatter plot points from different locations end up shown in the same location, around the contact wave one rather observes scatter than oscillations, i.e. a deviation from radial symmetry.

Theorems & Definitions (15)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Definition A.1
• Theorem A.1
• Theorem A.2
• proof
• Remark A.1
• Theorem A.3