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Active flux for triangular meshes for compressible flows problems

Rémi Abgrall, Jianfang Lin, Yongle Liu

TL;DR

This work develops an Active Flux method on triangular meshes, combining boundary degrees of freedom with a cell average to solve hyperbolic conservation laws in multiple dimensions. It provides a discretization framework with high-order schemes, a low-order fallback, linear stability considerations, and MOOD-based nonlinear stabilization to handle discontinuities while preserving invariants. The method is validated on scalar and Euler equations, achieving third- and fourth-order accuracy and effective shock-capturing across a suite of canonical problems, with limited activation of first-order schemes in MOOD. The study extends prior Active Flux ideas to simplex elements, offering a path toward virtual finite elements for linear or nonlinear hyperbolic problems with non-regular solutions.

Abstract

In this article, we show how to construct a numerical method for solving hyperbolic problems, whether linear or nonlinear, using a continuous representation of the variables and their mean value in each triangular element. This type of approach has already been introduced by Roe, and others, in the multidimensional framework under the name of Active flux, see \cite{AF1,AF2,AF3,AF4,AF5}. Here, the presentation is more general and follows \cite{Abgrall_AF,BarzukowAbgrall}. { Various} examples show the good behavior of the method in both linear and nonlinear cases, including non-convex problems. The expected order of precision is obtained in both the linear and nonlinear cases. This work represents a step towards the development of methods in the spirit of virtual finite elements for linear or nonlinear hyperbolic problems, including the case where the solution is not regular.

Active flux for triangular meshes for compressible flows problems

TL;DR

This work develops an Active Flux method on triangular meshes, combining boundary degrees of freedom with a cell average to solve hyperbolic conservation laws in multiple dimensions. It provides a discretization framework with high-order schemes, a low-order fallback, linear stability considerations, and MOOD-based nonlinear stabilization to handle discontinuities while preserving invariants. The method is validated on scalar and Euler equations, achieving third- and fourth-order accuracy and effective shock-capturing across a suite of canonical problems, with limited activation of first-order schemes in MOOD. The study extends prior Active Flux ideas to simplex elements, offering a path toward virtual finite elements for linear or nonlinear hyperbolic problems with non-regular solutions.

Abstract

In this article, we show how to construct a numerical method for solving hyperbolic problems, whether linear or nonlinear, using a continuous representation of the variables and their mean value in each triangular element. This type of approach has already been introduced by Roe, and others, in the multidimensional framework under the name of Active flux, see \cite{AF1,AF2,AF3,AF4,AF5}. Here, the presentation is more general and follows \cite{Abgrall_AF,BarzukowAbgrall}. { Various} examples show the good behavior of the method in both linear and nonlinear cases, including non-convex problems. The expected order of precision is obtained in both the linear and nonlinear cases. This work represents a step towards the development of methods in the spirit of virtual finite elements for linear or nonlinear hyperbolic problems, including the case where the solution is not regular.
Paper Structure (23 sections, 47 equations, 16 figures, 6 tables)

This paper contains 23 sections, 47 equations, 16 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Approximation space
  3. Numerical schemes
  4. High order schemes
  5. Low order schemes
  6. Linear stability
  7. Non-linear stabilization
  8. Numerical results
  9. Scalar case
  10. Convection problem
  11. Zalesak test case
  12. KPP test case
  13. Euler equations
  14. Accuracy test
  15. Moving vortex problem
  16. ...and 8 more sections

Figures (16)

  • Figure 1: Normals for \ref{['scheme:pt']}. Left: quadratic; Right: cubic.
  • Figure 2: Geometry for the first-order scheme. Left: quadratic; Right: cubic.
  • Figure 3: Zalesak problem: quadratic approximation and MOOD criteria are checked expect for PAD. Top row: $T=1$; Middle row: $T=2$; Bottom row: $T=3$.
  • Figure 4: Zalesak problem: quadratic approximation and all MOOD criteria are checked. Top row: $T=1$; Middle row: $T=2$; Bottom row: $T=3$.
  • Figure 5: Zalesak problem: cubic approximation and all MOOD criteria are checked. Top row: $T=1$; Middle row: $T=2$; Bottom row: $T=3$.
  • ...and 11 more figures