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High-Order Discontinuous Cut Finite Element Methods for Scalar Hyperbolic Conservation Laws

Pei Fu, Gunilla Kreiss, Zelin Xin, Sara Zahedi

arXiv:2609.08880v1math.NAcs.NA

TL;DR

Small cut cells in unfitted meshes can restrict explicit time steps, motivating robust high-order discretizations for complex domains. The paper combines CutFEM-DG with macro-element stabilization and adapted limiters. It reports maximum-principle preservation, expected convergence rates, oscillation-free shock solutions, and semi-discrete L2 stability.

  • Problem

    Small cut cells in background meshes can cause severe explicit time-step restrictions and ill-conditioned systems, complicating numerical methods for conservation laws on complex domains.

    Source: §1 · p. 3

  • Method

    The paper develops a high-order CutFEM-DG scheme using macro-elements, stabilization, and adapted flux, bound-preserving, and slope limiters.

    Source: §1 · p. 3, §2.4 · pp. 9–10

  • Results

    Experiments in two and three dimensions demonstrate expected convergence rates, maximum-principle preservation, and oscillation-free solutions near shocks or discontinuities.

    Source: §1 · p. 4

  • Takeaways & Limitations

    The proposed scheme combines maximum-principle preservation and shock handling with energy stability results for complex-domain scalar conservation laws.

    Source: §1 · p. 4, §5 · p. 26

    The analysis assumes a consistent, two-point Lipschitz numerical flux, and the macro-element construction assumes sufficiently small h so the large-intersection set is nonempty.

    Source: §2.1.1 · p. 6, §2.2 · p. 7

Abstract

from arXiv · show

In this paper, we present a family of high-order cut finite element methods based on the discontinuous Galerkin (DG) framework for scalar hyperbolic conservation laws on complex domains. Building on our previous work, we develop a multidimensional formulation that combines macro-element stabilization with flux limiting to obtain a scheme that preserves the maximum principle and remains robust with respect to arbitrary boundary cuts of the background mesh. The physical domain is embedded in a regular background mesh, which may produce arbitrarily small cut cells. To avoid the severe time step restrictions typically associated with such cells, ghost penalty stabilization terms are added on interior facets of macro-elements. The resulting method exhibits stability and accuracy properties similar to those of standard DG methods on fitted meshes. An $L^2$-stability result is derived for the semi-discrete scheme under both periodic and inflow-outflow boundary conditions. To enforce the maximum principle and suppress nonphysical oscillations, we adapt limiter techniques from standard DG methods to the CutFEM setting by defining limiting parameters on macro-elements. In particular, we present a macro-element-based parameterized flux limiter together with adaptations of the Zhang-Shu bound-preserving limiter and the Barth-Jespersen slope limiter. Numerical experiments in two and three spatial dimensions demonstrate optimal convergence orders, preservation of the maximum principle, and accurate shock capturing without spurious oscillations, even for challenging cut configurations involving very small-cut cell intersections.

1 Introduction

The paper develops a multidimensional CutFEM-DG method for scalar conservation laws on complex domains, where small cut cells challenge robust explicit discretization. Macro-element stabilization and limiting strategies address these challenges while targeting maximum-principle preservation and accurate shock handling.

1 Introduction

  • Small cut elements can impose severe time-step restrictions in explicit schemes and lead to ill-conditioned systems.These difficulties arise when only a very small portion of an element intersects the physical domain.Source: §1 · p. 3
  • The method extends CutFEM-DG to multiple dimensions and combines explicit Runge–Kutta time stepping with bound-preserving and slope limiters.The paper notes that arbitrary boundary cuts can produce complex macro-elements, making the higher-dimensional extension analytically nontrivial.Source: §1 · p. 3
  • The macro-element flux limiter controls mean values without computing exact polynomial extrema or requiring a cell-average decomposition on general macro-elements.A bound-preserving limiter is then applied at the final Runge–Kutta stage to constrain the polynomial solution; the adapted Barth-Jespersen slope limiter is applied at every stage near discontinuities.Source: §1 · p. 4
  • Numerical experiments in two and three dimensions show expected convergence rates, maximum-principle preservation, and oscillation-free solutions near shocks or discontinuities.The paper also derives energy stability results for the proposed scheme.Source: §1 · p. 4

2 The numerical scheme

The numerical scheme uses a discontinuous Galerkin discretization on an unfitted background mesh, organized into macro-elements. Stabilization and macro-element-based flux correction are paired with Runge–Kutta time stepping and adapted limiters.

2.1 Mesh and finite element space

  • The active mesh uses piecewise-polynomial finite element functions on background-mesh elements, with discontinuities across interior facets.The physical domain is embedded in a Cartesian background mesh.Source: §2.1 · p. 5

2.1.1 Macro-elements

  • Macro-elements group a root cell with nearby elements having small domain intersections, and may be constructed to have comparable size and contain only a few neighbors when the geometry is sufficiently smooth and resolved.The construction assumes a sufficiently small mesh size so that the set of large-intersection elements is nonempty.Source: §2.1.1 · p. 6

2.2 Semi-discrete unfitted DG method

  • The semi-discrete DG formulation uses a consistent, two-point Lipschitz numerical flux and stabilization terms on interior facets of nontrivial macro-elements.For inflow boundaries, the generalized Lax–Friedrichs parameter is taken greater than one for the stated stability analysis.Source: §2.2 · pp. 6–8, §2.1.1 · p. 6

2.3 Basic fully-discrete scheme

  • The fully discrete scheme advances the semi-discrete formulation with forward Euler updates, combined in the experiments using a third-order SSP Runge–Kutta method.The SSP-RK method is written as a convex combination of forward Euler updates.Source: §2.3 · p. 8

2.4 Flux-limiting

  • The method modifies numerical fluxes on macro-element facets to control macro-element mean values and preserve the maximum principle.The bound-preserving correction is applied after the final Runge–Kutta stage, and the flux limiter excludes interior facets where ghost penalty stabilization is applied.Source: §2.4 · pp. 9–10
  • The flux-limiting procedure compares high- and low-order fluxes and selects facetwise parameters to keep macro-element means within the maximum-principle bounds.For higher-order polynomial spaces, the flux correction is extended to the Runge–Kutta flux and applied after the third stage.Source: §2.4 · p. 10

11 end

The final listed algorithm steps loop over macro-elements and compute the corrected solution.

11 end

  • For each macro-element, the algorithm computes the corrected solution using the stated correction procedure.These steps specify the loop and correction computation, without further detail on the correction itself.Source: §11 · p. 11

17 end

The method selects macro-element flux-limiter parameters to preserve mean-value bounds while retaining as much high-order accuracy as allowed. It supplements mean control with slope and polynomial bound limiters, combined in a fully discrete Runge–Kutta procedure.

2.4.1 Parameters in the flux limiting

  • Limiter parameters are chosen to keep updated macro-element means within the prescribed bounds while remaining as close to 1 as the constraints permit.A shared facet parameter accounts for both neighboring macro-elements.Source: §2.4.1 · pp. 11–12
  • With θξ = 0, the low-order update preserves the maximum principle when the previous solution is piecewise constant and its macro-element means satisfy the bounds.For polynomial degree r ≥ 1, this choice yields the low-order update.Source: §2.4.1 · p. 11

2.5.1 Slope limiter

  • The adapted Barth–Jespersen limiter scales the polynomial deviation from each macro-element average to control oscillations near discontinuities.The scale is determined using the macro-element and neighboring macro-elements sharing a vertex, together with quadrature points on its sub-elements.Source: §2.5.1 · p. 13

2.5.2 Bound-preserving limiter

  • The Zhang–Shu bound-preserving limiter controls the polynomial solution on each macro-element at the final stage of every Runge–Kutta step.The flux limiter bounds macro-element means, but polynomial values and element-wise means may still violate those bounds.Source: §2.5.2 · p. 13

2.6 Fully discrete maximum-principle-preserving scheme

  • The fully discrete procedure applies the flux limiter after the third SSP-RK stage, then uses slope limiting as needed and finishes with polynomial bound-preserving limiting.Slope limiting may also be applied after the first two stages for problems with discontinuities.Source: §2.6 · p. 14

3 Main analytical results

The analysis establishes stability and maximum-principle properties for the CutDG scheme, along with an a priori error estimate for linear advection. The bounds apply under stated assumptions and do not deteriorate with the cut configuration.

3.1 L2-stability of the semi-discrete scheme

  • The semi-discrete scheme is L2-stable for periodic boundaries and, under the stated generalized Lax–Friedrichs condition, inflow-outflow boundaries; its estimate is independent of cut configuration.With inflow boundaries, the stabilized energy is bounded by initial energy and prescribed boundary data.Source: §3.1 · pp. 15, 17
  • For linear advection, the paper derives an a priori error estimate whose constant is independent of the mesh parameter and interface-mesh intersection.The analysis assumes the exact solution and its time derivative have the stated Sobolev regularity.Source: §3.1 · p. 18

3.2 The fully discretized scheme

  • The fully discretized analysis establishes maximum-principle preservation for the low-order piecewise-constant scheme and the high-order scheme's macro-element means.The high-order result uses the proposed flux-limiter parameters.Source: §3.2 · p. 18, §3.2.1 · p. 18, §3.2.2 · p. 20

3.2.1 Low order scheme

  • The low-order macro-element mean update preserves the maximum principle under a CFL condition that scales as in the fitted case, avoiding severe time-step restrictions from arbitrarily small cut cells.The result assumes that the previous macro-element means satisfy the bounds.Source: §3.2.1 · pp. 18–20

3.2.2 High order scheme

  • When the low-order theorem's assumptions hold, the flux-limited high-order update also keeps macro-element mean values within the prescribed bounds.The facet parameters are chosen to satisfy the admissibility condition on neighboring macro-elements simultaneously.Source: §3.2.2 · pp. 20–21

4 Numerical examples

The experiments use uniformly refined Cartesian background meshes, SSP-RK time stepping, and a time step proportional to h^α, with Dirichlet inflow and extrapolated outflow data. Across smooth linear advection tests in two and three dimensions, the method shows optimal convergence; discontinuous advection and Burgers tests demonstrate maximum-principle preservation and shock capture.

4.1.1 A two dimensional problem

  • For 2D smooth linear advection, MPP flux limiting avoids macro-element mean overshoots and undershoots, and the errors indicate optimal convergence.Figure 1 shows the mesh, a P 3 solution at h = 1/128, and L2 and L∞ errors.Source: §4.1.1 · p. 22, Figure 1 · p. 22

4.1 Linear problems with constant coefficients

  • For the 2D linear-advection equation, optimal convergence rates are reported; in 3D, optimal convergence is observed for both P1 and P2, with no obvious oscillations in the displayed P2 solution.With MPP flux limiting applied in the 2D test, no overshoots or undershoots of macro-element mean values are observed.Source: §4.1.1 · p. 22, Figure 1 · p. 22, §4.1.2 · p. 22

4.1.2 A three dimensional problem

  • In 3D linear advection, the P 2 solution shows no obvious oscillations, and both P 1 and P 2 exhibit optimal convergence.Figure 2 displays the P 2 solution at h = 2.2/80 and L2 and L∞ errors.Source: §4.1.2 · p. 22, Figure 2 · p. 22

4.1.3 A two dimensional non-smooth problem

  • For nonsmooth 2D linear advection, the method captures discontinuities while keeping macro-element mean values within the admissible bounds up to machine precision.Without the MPP flux limiter, oscillations are largely suppressed but the maximum principle is no longer guaranteed.Source: §4.1.3 · p. 23, Figure 3 · p. 23, Figure 4 · p. 24

4.2 Nonlinear problems: Burgers’ equation

  • For smooth Burgers’ equation before shock formation, the CutDG method achieves optimal accuracy for P 1, P 2, and P 3 approximations.The errors shown are L2 and L∞ at T = 0.5/π.Source: §4.2 · p. 24, Figure 5 · p. 24
  • For Burgers’ equation after shock formation, the method captures shocks while maintaining the maximum principle; applying all three limiters also bounds Gaussian quadrature-point values but makes the method slightly more diffusive.With flux splitting, the method retains expected optimal convergence and macro-element mean MPP, and gives nearly identical solutions to the original formulation.Source: §4.2 · pp. 24–25, Figure 6 · p. 25, Figure 8 · p. 26

5 Conclusion

The proposed CutDG methods preserve the maximum principle and suppress spurious oscillations, with optimal convergence for smooth solutions and robust shock capturing for discontinuous problems. Their time-step restriction is comparable to fitted DG and essentially independent of cut configuration.

5 Conclusion

  • Numerical experiments showed optimal convergence for smooth solutions and robust shock capturing for discontinuous problems.The time-step restriction remains comparable to fitted DG and is essentially independent of cut configuration.Source: §5 · p. 26
  • Macro-element flux limiting and adapted Zhang-Shu and Barth–Jespersen limiters preserve the maximum principle and suppress spurious oscillations near discontinuities.Source: §5 · p. 26
  • The semi-discrete scheme is L2-stable under periodic and inflow–outflow boundary conditions.Source: §5 · p. 26
  • The framework is also applicable to systems of conservation laws, while extension to the Euler equations remains ongoing work.Source: §5 · p. 26
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