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On the involution preserving properties of semi-discrete Active Flux methods on simplex

Remi Abgrall, Yongle Liu, Mario Ricchiuto

arXiv:2610.04349v1math.NAcs.NA

TL;DR

The paper asks how a discretization can satisfy both conservative form and the involution constraints of acoustics and Maxwell equations. It derives discrete curl and divergence operators preserved by Active Flux on simplex meshes. The analysis covers arbitrary polynomial degree, and third-order experiments support the theoretical results.

  • Problem

    The paper addresses how a single discretization can satisfy conservative form and the corresponding involution for linear acoustics and Maxwell equations.

    Source: §1.1 · p. 3

  • Method

    The authors analyze discrete Active Flux on unstructured meshes and derive explicit discrete curl and divergence operators for the two systems.

    Source: §1.3 · p. 3

  • Results

    The scheme preserves the derived discrete involutions, and third-order experiments report pointwise curl preservation for acoustics and very well-preserved divergence for Maxwell.

    Source: §5.1.2 · p. 12, §5.2.1 · p. 13, §6 · p. 14

  • Takeaways & Limitations

    The analysis applies to simplex meshes and any polynomial degree, including fully discrete schemes when explicit time stepping is used.

    Source: §1.3 · p. 3, §6 · p. 14

    The analysis does not apply as stated to quadrangular or polygonal meshes, and it leaves unresolved whether discrete gradient or curl-potential initial data remain so over time.

    Source: §2 · p. 4, §6 · p. 14

Abstract

from arXiv · show

We study the involution-preserving properties of the Active Flux method on unstructured triangular meshes. Two model problems are considered: the two-dimensional linear acoustic system, which satisfies a curl involution on the velocity field, and the two-dimensional Maxwell equations with an in-plane magnetic field, which satisfy a divergence involution. Both systems can be written in conservative form. The question we address is how a single discretization can satisfy the conservative form and the involution at the same time. Within a discrete Active Flux framework on simplices, we derive explicit formulas for a discrete curl operator that is exactly preserved by the scheme for acoustics, and for a discrete divergence operator preserved for the Maxwell system. The analysis applies to general simplex meshes and to any polynomial degree, including extensions of the method beyond third order. Numerical experiments with the third-order scheme confirm the theoretical results. Highlights: 1-Active Flux on unstructured triangular meshes preserves discrete involutions, 2-Explicit discrete curl operator preserved by the scheme for linear acoustics, 3- Explicit discrete divergence operator preserved for 2D Maxwell equations, 4- Results hold on general simplex meshes,% and for any polynomial degree 5- Third-order numerical experiments confirm the theoretical results. \end{itemize}

1 Introduction

The paper asks how one discretization can preserve both conservative form and involution for hyperbolic systems. It studies curl preservation for acoustics and divergence preservation for Maxwell equations, deriving explicit discrete operators for Active Flux on unstructured meshes.

1.1 PDE systems and involutions

  • The acoustic system has a curl involution, while the Maxwell system with an in-plane magnetic field has a divergence involution.The paper relates these constraints to div curl = 0 and rot grad = 0.Source: §1.1 · p. 2, §1 · p. 2
  • The central question is how to design methods that satisfy both the conservative form and the corresponding curl or divergence involution.The systems considered are two-dimensional linear acoustics and Maxwell equations.Source: §1.1 · p. 3

1.2 A (partial) state of the art

  • For unstructured Active Flux, prior work suggested curl compatibility, but an explicit discrete curl definition was not known.Earlier Active Flux work had shown stationary-preserving features linked to involution preservation in linear acoustics.Source: §1.2 · p. 3

1.3 Contribution of this paper

  • The paper gives explicit discrete curl and divergence operators preserved by Active Flux, with theory for simplex meshes and any approximation degree.Numerical results are provided for the third-order scheme.Source: §1.3 · p. 3

2 Approximation: triangular case

The approximation is built on regular triangular meshes using continuous polynomial representations with nodal and element-average degrees of freedom. For higher degrees, the construction adds edge points and moments while retaining the polynomial space needed for accuracy.

2 Approximation: triangular case

  • The mesh consists of nonoverlapping polygons, with the analysis focused on regular triangular meshes and consistently oriented edge tangents and normals.The text notes that general cases receive comments elsewhere.Source: §2 · p. 4
  • For linear and quadratic approximations, globally continuous polynomials use vertex values, quadratic edge-midpoint values, and an element average as degrees of freedom.The average is represented by a bubble function vanishing on the element boundary.Source: §2 · p. 4

2.1 Higher order extension

  • For degree k > 2, the approximation uses vertices, k−1 Gauss–Lobatto points per edge, and k(k−1)/2 moments, giving local dimension 3k+k(k−1)/2.The space is designed to be globally continuous and to contain Pk.Source: §2.1 · p. 4
  • The higher-order interpolation forms are stated to be unisolvent on the approximation space.The cubic triangular case's interpolation points are illustrated in Figure 1.Source: §2.1 · p. 5, Figure 1 · p. 6

3 Discretization of the PDEs

The discretization updates element averages, point values, and higher-order moments for the acoustic and Maxwell systems. Its higher-order extension evaluates boundary contributions from degree-k traces and uses quadrature for surface integrals.

3.1 Third order scheme

  • For the third-order scheme, the element-average update evaluates the boundary integral exactly with Simpson's formula on each edge.The boundary trace is quadratic and known.Source: §3.1 · p. 6
  • Point values are updated following the Active Flux procedure in, with edge degrees of freedom using the outward normal of their containing side.The normals are element-dependent; the implementation typically sets ε to zero except at boundary points.Source: §3.1 · pp. 6–7

3.2 Higher order extensions

  • For higher degrees, moments are updated using computable boundary integrals and quadrature for surface integrals; Gauss–Lobatto point values follow the quadratic-case procedure in.The boundary trace has degree k.Source: §3.2 · p. 7

4 Discrete involutions

For triangular Active Flux, the acoustics construction yields curl involutions associated with vertices and, at higher degree, edge moments; a parallel construction gives Maxwell div involutions.

4.1 Acoustics and nodal curl involutions

  • The acoustic scheme has one curl involution per vertex; the proof uses the piecewise-constant gradient of a linear test function in the element average.The argument does not extend to edge midpoints by the same test-function reasoning.Source: §4.1 · p. 8
  • For k ≥ 3, the acoustics-system velocity is approximated by degree-k Active Flux, and gradients of degree-(k−2) moment polynomials serve as test functions.Source: §4.1 · pp. 8–9
  • For k>2 on a triangle, the acoustic involution count is 3 + 3(k−2): one per vertex plus k−2 per edge.The vertex-and-average argument also has stated limitations for polygons.Source: §4.1 · pp. 8–9
  • On tetrahedra, the acoustic count extends to vertices, degree-dependent edge terms, and degree-dependent face terms.The stated count is nv + max(k−2, 0)nedge + max(k−3, 0)nfaces, with four vertices, six edges, and four faces.Source: §4.1 · p. 9

4.2 Maxwell equations and nodal div involutions

  • For Maxwell, the construction gives a div involution at each vertex and extends to higher degree through relations between moments of different degrees.The weak-div construction is stated for the degree-k Active Flux magnetic-field approximation.Source: §4.2 · pp. 10–11
  • The Maxwell involution count follows the same rule as the acoustic count in both two and three dimensions.The paper relates this structure to Active Flux’s behavior for Maxwell.Source: §4.2 · p. 11

5 Numerical illustrations.

The numerical illustrations test acoustic curl preservation and Maxwell divergence preservation in steady and unsteady problems. The reported results include vertexwise invariance, recovery of expected order, and divergence preservation.

5.1.1 Steady vortex

  • For the steady vortex, vertex-evaluated vorticity remains invariant in time up to truncation errors.Figure 2 reports L1, L2, and L∞ norms of ROT(u − u0) evaluated at vertices.Source: §5.1.1 · p. 11, Figure 2 · p. 12

5.1.2 Unsteady vortex

  • In the reported results, the expected order is recovered, and the initial curl is well preserved pointwise, as illustrated in Figure 2-(b).Table 1 reports average-value errors for problem (18).Source: §5.1.2 · p. 12, Table 1 · p. 13

5.2.1 Steady problem

  • In the steady Maxwell problem, divergence is very well preserved; Figure 3 shows its L1, L2, and L∞ norms.The divergence is evaluated at vertices.Source: §5.2.1 · p. 13, Figure 3 · p. 13

5.2.2 Unsteady problem

  • The divergence-preservation test uses a coarse mesh of only 4029 points; its L1-norm error is reported in Table 2 and is described as expected.Source: §5.2.2 · p. 14

6 Conclusions and perspectives

The analysis gives explicit nodal discrete curl and divergence operators preserved by Active Flux for linear acoustics and Maxwell equations. Its scope excludes polygonal meshes as presented and leaves related preservation questions open.

6 Conclusions and perspectives

  • For triangular meshes in 2D and tetrahedral meshes in 3D, the method preserves curl-free or solenoidal structure and provides explicit nodal discrete curl and divergence operators.The result also applies to the fully discrete method when explicit time stepping is used.Source: §6 · p. 14
  • The analysis does not apply as presented to quadrangular or polygonal meshes, and it does not establish whether discrete solutions preserve gradient or curl-potential initial data.It also leaves open whether the fully discrete method discretizes the wave equation.Source: §6 · p. 14

A Proof of Lemma 2.1

The proof establishes unisolvence of the approximation forms by eliminating nodal coefficients and showing that the remaining moment matrix is positive definite.

A Proof of Lemma 2.1

  • Testing with Lagrange polynomials shows that the nodal coefficients ασ vanish.The proof first introduces Lagrange polynomials at the nodal points.Source: A Proof of Lemma 2.1 · p. 15
  • The remaining question is matrix invertibility; its positive definiteness establishes that the forms are unisolvent.The passage identifies the matrix as symmetric and of size |Ik| × |Ik|.Source: A Proof of Lemma 2.1 · p. 15

B Well posedness of the nodal weighting

The nodal-weighting analysis establishes invertibility through positive definiteness in element interiors, with a separate boundary treatment where invertibility is not guaranteed.

B Well posedness of the nodal weighting

  • The acoustics Jacobian has eigenvalues 0 and ±‖n‖, while replacing the normal by the tangent gives the same eigenvalues.The argument uses the normal or tangent direction in the corresponding Jacobian analysis.Source: B Well posedness of the nodal weighting · pp. 16–17
  • For interior points, the weighting matrix is symmetric positive definite and therefore invertible; boundary points may yield only a semidefinite matrix.The construction sets ε to zero at boundary points.Source: B Well posedness of the nodal weighting · p. 16
  • The tangential-direction argument also excludes nonzero null vectors, allowing ε to be set to zero except at boundary points.For Maxwell, the analysis proceeds similarly using tangent vectors.Source: B Well posedness of the nodal weighting · pp. 16–17
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