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New Fully Discrete Active Flux Methods with Truly Multi-Dimensional Evolution Operators and WENO Reconstruction

Amelie Porfetye, Zhuyan Tang, Shaoshuai Chu, Christiane Helzel, Maria Lukacova-Medvidova

TL;DR

This work advances fully discrete Active Flux methods for the 2D acoustic equations by introducing new multidimensional evolution operators. The EG^{quad} operator reproduces quadratic planar waves exactly, while EG2_{\delta} and EG2_{\delta,\nu} substantially improve stability, enabling larger CFL numbers; all variants maintain third-order accuracy on smooth problems. The authors augment AF with a Central CWENO reconstruction (AFCW) to further boost robustness, achieving high-order accuracy with a compact stencil and good behavior near discontinuities. Overall, the proposed operators extend the practical time-step range and preserve stability across test problems, with exact evolution uniquely preserving stationary vortices. Future work aims to generalize to nonlinear systems like the Euler equations and to investigate positivity-preserving and limiting properties of the AFCW framework.

Abstract

We propose new fully discrete third-order accurate Active Flux and WENO methods based on truly multidimensional evolution operators for the two-dimensional acoustic equations. Building on the method of bicharacteristics, several approximate evolution operators are derived that yield an improved stability of the resulting schemes. A linear stability analysis is applied to determine the maximal CFL number. The schemes are tested extensively on both continuous and discontinuous problems, confirming their robustness and accurate approximation even on coarse grids.

New Fully Discrete Active Flux Methods with Truly Multi-Dimensional Evolution Operators and WENO Reconstruction

TL;DR

This work advances fully discrete Active Flux methods for the 2D acoustic equations by introducing new multidimensional evolution operators. The EG^{quad} operator reproduces quadratic planar waves exactly, while EG2_{\delta} and EG2_{\delta,\nu} substantially improve stability, enabling larger CFL numbers; all variants maintain third-order accuracy on smooth problems. The authors augment AF with a Central CWENO reconstruction (AFCW) to further boost robustness, achieving high-order accuracy with a compact stencil and good behavior near discontinuities. Overall, the proposed operators extend the practical time-step range and preserve stability across test problems, with exact evolution uniquely preserving stationary vortices. Future work aims to generalize to nonlinear systems like the Euler equations and to investigate positivity-preserving and limiting properties of the AFCW framework.

Abstract

We propose new fully discrete third-order accurate Active Flux and WENO methods based on truly multidimensional evolution operators for the two-dimensional acoustic equations. Building on the method of bicharacteristics, several approximate evolution operators are derived that yield an improved stability of the resulting schemes. A linear stability analysis is applied to determine the maximal CFL number. The schemes are tested extensively on both continuous and discontinuous problems, confirming their robustness and accurate approximation even on coarse grids.

Paper Structure

This paper contains 22 sections, 2 theorems, 47 equations, 8 figures, 24 tables.

Table of Contents

  1. Introduction
  2. Fully discrete Cartesian Grid Active Flux methods
  3. Previously proposed truly multi-dimensional evolution operators
  4. Exact evolution operator
  5. EG2 approximate evolution operator
  6. Test problems
  7. New Evolution Operator Reproducing Exactly Quadratic Plane Waves
  8. New Evolution Operator with Increased Stability
  9. Investigation of linear stability
  10. Investigation of accuracy
  11. Approximation of the Stationary Vortex
  12. Approximation of discontinuous solutions
  13. Central Weighted Essentially Non-Oscillatory Reconstruction
  14. Investigation of accuracy
  15. Approximation of the Stationary Vortex
  16. ...and 7 more sections

Key Result

Lemma 4.1

Let $f \in C^3\left( \Omega\right)$, where $\Omega \subset \mathbb{R} ^2$ and that contains the closed disk $\overline{B_R\left( x_0,y_0\right) }$. For $R>0$, define the circle parametrisation $\mathbf{Q}_R\left( \theta\right) =\left( x_0+R\cos\left( \theta\right) , y_0+R\sin\left( \theta\right) \ri for $R > 0$ sufficiently small.

Figures (8)

  • Figure 1: Eigenvalues of the matrix $B$ for the AF method using (from left to right): exact evolution, EG2, $\mathrm{EG}^{\text{quad}}$, EG2$_{0.7}$ and EG2$_{0.8,0.2}$ with CFL $=0.44$. A $20\times 20$ grid with periodic boundary conditions was used.
  • Figure 2: Eigenvalues of the matrix $B$ for the AF method using exact evolution, EG2 and $\mathrm{EG}^{\text{quad}}$ with CFL $=0.279$. A $20\times 20$ grid with periodic boundary conditions was used.
  • Figure 3: Approximation of the stationary vortex using a grid with $64\times64$ (top) and $128\times128$ (bottom) cells at $t=100$ with an AF method using exact evolution(left), $\mathrm{EG}^{\text{quad}}$ (center left), EG2$_{0.7}$ (center right) and EG2$_{0.8,0.2}$ (right).
  • Figure 4: Approximation of discontinuous solution at $t=0.5$ on grids with $64\times 64$ (left), $128\times128$ (middle) and $256\times 256$ (right) cells using the AF method with exact evolution (first row), EG2 (second row), EG2$_{0.7}$ (third row) and $\mathrm{EG}^{\text{quad}}$ (fourth row). All methods use time steps near their stability limit.
  • Figure 5: Approximation of the stationary vortex using a grid with $64 \times 64$ (top) and $128 \times 128$ (bottom) cells at $t=100$ with AFCW method using EG2 (left), $\mathrm{EG}^{\text{quad}}$ (middle) and EG2$_{0.8,0.2}$ (right) operators, respectively.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 4.1
  • proof
  • Remark 4.1
  • Lemma A.1
  • proof