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Invariant domains and first-order continuous finite element approximation for hyperbolic systems

Jean-Luc Guermond, Bojan Popov

arXiv:1509.07461v1math.NAcs.NA

TL;DR

General hyperbolic systems need a first-order approximation that controls invariant sets and entropy behavior. The paper introduces a continuous finite element method with artificial dissipation and proves invariant-domain preservation under a CFL condition. It also establishes a discrete entropy inequality, while noting that invariant-domain preservation alone does not ensure convergence.

  • Problem

    The paper targets first-order approximation of general hyperbolic systems using continuous finite elements while preserving convex invariant sets and entropy behavior.

    Source: §1 · pp. 1–2

  • Method

    The method uses forward Euler time stepping, continuous finite elements, and artificial dissipation chosen to preserve convex invariant sets under a CFL condition.

    Source: §1 · p. 1, §3.3 · p. 11, §6 · p. 21

  • Results

    The scheme preserves convex invariant sets and satisfies a discrete entropy inequality; the paper also shows that invariant-domain preservation alone does not imply convergence.

    Source: §1 · p. 2, §5.1 · p. 17

  • Takeaways & Limitations

    For the illustrated convergence issue, local entropy inequalities are essential in addition to invariant-domain preservation.

    Source: §5.1 · p. 17

    The discussed speed estimate can be insufficient for expansion or composite waves, which may arise with nonconvex fluxes.

    Source: §5.1 · p. 16

Abstract

from arXiv · show

We propose a numerical method to solve general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on non-uniform grids. The properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant sets containing the initial data is an invariant domain for the method. The invariant domain property is proved for any hyperbolic system provided a CFL condition holds. The solution is also shown to satisfy a discrete entropy inequality for every admissible entropy of the system. The method is formally first-order accurate in space and can be made high-order in time by using Strong Stability Preserving algorithms. This technique extends to continuous finite elements the work of \cite{Hoff_1979,Hoff_1985}, and \cite{Frid_2001}.

1. Introduction.

The introduction frames a first-order continuous finite element approximation for hyperbolic systems and presents invariant-domain preservation and discrete entropy inequalities as its main results.

1. Introduction.

  • The proposed method uses explicit time stepping and continuous finite elements on non-uniform grids in any space dimension.The authors describe it as a loose adaptation of the non-staggered Lax-Friedrichs scheme to continuous finite elements.Source: §1 · p. 1
  • The method preserves every convex invariant set and satisfies a discrete entropy inequality for every entropy pair, with no system restriction beyond finite propagation speed.The results are stated in Theorems 4.1 and 4.2; the authors report no similar scheme in the continuous finite element literature.Source: §1 · p. 2
  • The paper introduces invariant sets and invariant domains, presents the method, analyzes its stability, and gives numerical illustrations and comparisons.The introduction notes that its definitions differ slightly from earlier treatments.Source: §1 · p. 2, §2 · p. 2

2. Preliminaries.

The preliminaries set assumptions on Riemann solutions, distinguish invariant sets from invariant domains, and illustrate them for scalar equations, the p-system, and Euler equations.

2.1. Riemann problem.

  • Finite propagation speed bounds the Riemann solution’s domain of influence; under the stated CFL bound, its averages support invariant-set and entropy arguments.The Riemann solution is assumed to have finite propagation speed, and an entropy pair is used when the Riemann solution satisfies the entropy condition.Source: §2.1 · p. 3

2.2. Invariant sets and domains.

  • An invariant set constrains Riemann-solution averages, whereas an invariant domain requires the discrete process to keep states within a convex invariant set.The paper’s objective is a continuous finite element method for which every convex invariant set is an invariant domain.Source: §2.2 · p. 4

2.3. Example 1: scalar equations.

  • For scalar conservation equations, bounded intervals are invariant sets, and the corresponding numerical maximum principle keeps solutions within those intervals.The stated admissibility condition includes a Lipschitz flux.Source: §2.3 · pp. 4–5

2.4. Example 2: p-system.

  • For the p-system, the paper describes invariant regions and develops an upper-bound estimate for the maximum wave speed, including cases with vacuum formation.The cited invariant-region example uses the gamma-law pressure; the wave-speed estimate is stated to apply whether or not vacuum forms.Source: §2.4 · pp. 5–6

2.5. Example 3: Euler.

  • For Euler equations, a change to normal and tangential coordinates reduces the density, normal velocity, and pressure part of the Riemann problem to a one-dimensional problem.The Jacobian does not depend on transverse momentum; the discussion assumes positive density and nonnegative pressure in the non-vacuum states.Source: §2.5 · p. 7

3. First order method.

The method combines continuous finite element spaces with an explicit update and artificial viscosity; its convex-combination argument supplies the stated stability mechanism.

3.1. The finite element space.

  • The finite element space is continuous and uses Lagrange elements on shape-regular affine matching meshes, with an elementwise nodal bound assumption.The text lists first-order simplex, parallelogram, cuboid, and prismatic elements as examples satisfying the assumption.Source: §3.1 · pp. 8–9

3.2. The scheme.

  • The explicit update uses a lumped mass matrix for the time derivative and pairwise artificial-viscosity coefficients.The coefficients are assumed symmetric and nonnegative off the diagonal.Source: §3.2 · p. 10

3.3. The convex combination argument.

  • With a sufficiently small time step, the update can be written as a convex combination, and the selected viscosity relates intermediate states to Riemann-solution averages.The viscosity choice is motivated by a CFL bound involving the maximum propagation speed and the mesh-associated coefficient.Source: §3.3 · p. 11
  • In the scalar one-dimensional linear-flux case, the scheme reduces to the usual first-order upwinding method.Source: §3.3 · p. 11

4. Stability analysis.

The stability analysis shows that, under a CFL condition, the scheme preserves convex invariant sets and satisfies a discrete entropy inequality. It also examines cell-based viscosity as an alternative that retains these properties without relying on the standard Laplacian form’s mesh-shape robustness.

4. Stability analysis.

  • Using the local minimum mesh size instead of the global minimum sharpens the CFL-number estimate.The mesh quantities are uniformly equivalent under the shape-regularity assumption.Source: §4.1 · p. 11

4.1. Invariant domain property.

  • Under a CFL condition and bounded maximum wave speed, the scheme keeps its solution in any convex invariant set containing the initial data.The theorem also gives a local invariance result for a smaller convex invariant set containing the relevant neighboring states.Source: §4.1 · pp. 11–12

4.2. Discrete entropy inequality.

  • Under the stated Riemann-entropy and CFL assumptions, the scheme satisfies a local discrete entropy inequality, which also yields a global entropy inequality.The authors note that the proof needs only condition (2.8), which could replace the potentially ambiguous condition (2.7).Source: §4.2 · pp. 12–13

4.2.1. Cell-based vs.

  • The cell-based viscosity formulation retains the invariant-domain and entropy conclusions under the theorem’s assumptions.It introduces a piecewise-constant viscosity field through the weak form of the Laplacian.Source: §4.2.1 · pp. 13–14
  • The standard Laplacian bilinear form is not robust to cell shape; the proposed alternative supports the convex-combination argument independently of space dimension and mesh-family shape regularity.The replacement form is introduced to avoid the acute-angle restriction associated with the standard form.Source: §4.2.1 · pp. 13–14

5. Numerical illustrations.

The numerical examples show that preserving an invariant domain or maximum principle alone does not ensure convergence to the entropy solution. The proposed method’s entropy condition and invariant-domain property are illustrated alongside non-monotone but admissible profiles.

5. Numerical illustrations.

  • The counterexample tests whether a formally first-order, invariant-domain-preserving method necessarily converges.The authors frame it as a counterexample to that implication.Source: §5 · p. 15

5.1. Invariant domain property and convergence issues.

  • The algebraic diffusion choice can satisfy the maximum principle yet be insufficient for general fluxes, because average wave speed may not bound expansion or composite waves adequately.The paper notes that this wave-speed estimate is correct for simple shocks but may be insufficient for expansion or composite waves, especially with non-convex fluxes.Source: §5.1 · p. 16
  • The KPP comparison shows that a maximum-principle-satisfying solution can still be entropy-violating, while the proposed diffusion yields the entropy-satisfying composite wave.The algebraic method’s artificial viscosity is insufficient for this non-convex flux; the authors conclude that local entropy inequalities are essential for convergence.Source: §5.1 · p. 17, Figure 5.1 · p. 17
  • Satisfying the invariant-domain property or maximum principle does not by itself imply convergence, even for a formally first-order method.The paper contrasts the entropy-satisfying scheme with the algebraic method, which lacks the local entropy inequalities cited as essential for convergence.Source: §5.1 · p. 17

5.2. Special meshes.

  • On symmetric interior nodes, the special-mesh construction pairs diffusion coefficients; elsewhere it uses a lower bound, and the resulting diffusion matrix is conservative.The paired construction relies on the symmetry property; the fallback bound is used when that property does not hold.Source: §5.2 · p. 18
  • Invariant-domain preservation and what the literature calls monotonicity are distinct concepts, so monotonicity alone can mislead.The numerical profiles illustrate this distinction in the p-system and Leblanc shocktube examples.Source: §5.3 · p. 18, §5.5 · p. 21

5.4. p-system.

  • For the p-system, the method converges with order close to 0.9, while its single-rarefaction profile has a mesh-refining overshoot at the wave foot.The overshoot is not a Gibbs phenomenon; the paper describes it as a start-up error.Source: §5.4 · pp. 19–20

5.5. Euler in 1D (Leblanc shocktube).

  • The Leblanc shocktube density profile has a small dip at the rarefaction foot despite positivity and invariant-domain preservation, showing that invariance differs from monotonicity.The numerical solution remains in the smallest convex invariant set containing the Riemann data.Source: §5.5 · pp. 20–21
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