An Energy-Preserving Domain of Dependence Stabilization for the Linear Wave Equation on Cut-Cell Meshes

We present an energy-preserving (either energy-conservative or energy-dissipative) domain of dependence stabilization method for the linear wave equation on cut-cell meshes. Our scheme is based on a standard discontinuous Galerkin discretization in space and an explicit (strong stability preserving) Runge Kutta method in time. Tailored stabilization terms allow for selecting the time step length based on the size of the background cells rather than the small cut cells by propagating information across small cut cells. The stabilization terms preserve the energy stability or energy conservation property of the underlying discontinuous Galerkin space discretization. Numerical results display the high accuracy and stability properties of our scheme.

💡 Research Summary

The paper addresses the notorious “small‑cell problem” that arises when using cut‑cell meshes for explicit time integration of hyperbolic PDEs, specifically the linear acoustic wave equation in first‑order form. In cut‑cell meshes, cells intersected by an embedded geometry can become arbitrarily small, forcing the Courant–Friedrichs–Lewy (CFL) condition to be dictated by the smallest cell size. This leads to prohibitively small time steps for explicit schemes, undermining the main advantage of cut‑cell methods—easy mesh generation for complex geometries.

The authors propose a novel stabilization technique based on the Domain‑of‑Dependence (DoD) concept, originally introduced for linear advection, and adapt it to the linear wave system while preserving the energy properties of the underlying discontinuous Galerkin (DG) discretization. The base spatial discretization is a standard DG method of arbitrary polynomial order r on a Cartesian background mesh. The time discretization is an explicit strong‑stability‑preserving (SSP) Runge–Kutta scheme.

Key ingredients of the method are:

1. Cell classification – each cut cell E is assigned a volume fraction α_E = |E|/|Ē|, where Ē is the corresponding background cell. Cells with α_E below a user‑defined threshold α are marked as “small” and collected in the set I. The authors assume (i) small cells do not share a face with another small cell, (ii) any boundary face of a small cell lies on a planar portion of the domain boundary (constant normal), and (iii) a geometric regularity condition linking cell diameter and volume. These assumptions simplify the construction but are not fundamentally required for the theory.

2. Mirroring operator – to enforce reflecting (v·n = 0) boundary conditions weakly, a mirroring operator M_n is defined. For a state u = (p, v)ᵀ, M_n(u) = (p, v – 2(v·n)n)ᵀ. This operator is skew‑symmetric with respect to the flux matrix A_n, guaranteeing that the boundary contribution does not create or destroy energy.

3. Splitting of numerical fluxes – the numerical flux H_n is split into a central part (average of physical fluxes) and a dissipative part (e.g., Lax–Friedrichs). This split is essential because the central part admits an integration‑by‑parts identity that can be used to construct “propagation forms”.

4. Propagation forms – two families of bilinear forms are introduced: one for the central flux (J₀) and one for the dissipative flux (J₁). Each form satisfies a discrete integration‑by‑parts property that mimics the continuous energy balance. For a small cell E ∈ I, the stabilization terms J₀,E and J₁,E couple the solution inside E with its neighboring large cells, effectively allowing information to “propagate” across the small cell within a single time step.

5. Stabilized semi‑discrete system – the final semi‑discrete equation reads
   ⟨∂_t u_h, w_h⟩ + a_h(u_h, w_h) + J_a_h(u_h, w_h) + s_h(u_h, w_h) + J_s_h(u_h, w_h) = 0,
   where a_h and s_h are the standard DG volume and surface contributions, while J_a_h and J_s_h are the sums of the propagation forms over all small cells.

The authors prove that the stabilized scheme inherits the energy property of the underlying DG method: if only the central flux is used, the scheme is exactly energy‑conservative; if the dissipative flux is added, the scheme is energy‑dissipative with a controllable amount of dissipation determined by the Lax–Friedrichs coefficient. Crucially, the CFL condition for the explicit Runge–Kutta integrator depends only on the background mesh size h, not on the minimal cut‑cell volume.

Numerical experiments are performed on two‑dimensional domains with circular and more intricate embedded interfaces, generating cut‑cell meshes with a wide range of cell sizes. Polynomial orders r = 1, 2, 3 are tested. The results demonstrate:

• Stability – even with a time step chosen according to the background CFL (e.g., CFL ≈ 0.5), the solution remains stable and no spurious overshoots appear near tiny cut cells.
• Energy behavior – for the energy‑conservative variant, the discrete energy remains constant up to machine precision over many thousands of time steps; for the dissipative variant, the energy decays monotonically as predicted.
• Convergence – L² error norms exhibit the expected (r + 1)‑th order convergence, confirming that the stabilization does not degrade the high‑order accuracy of the DG method.
• Boundary handling – reflecting boundaries are treated weakly via the mirroring operator, and the numerical results confirm that the energy balance holds exactly at the walls.

Overall, the paper delivers a clean, theoretically sound, and practically implementable framework for explicit high‑order DG simulations on cut‑cell meshes. By separating central and dissipative fluxes and constructing propagation forms that respect a discrete integration‑by‑parts identity, the method achieves two goals simultaneously: (i) removal of the restrictive small‑cell CFL constraint, and (ii) preservation of the energy structure of the continuous wave equation. The authors suggest future extensions to nonlinear wave systems, three‑dimensional geometries, and relaxation of the small‑cell adjacency assumption, indicating a broad potential impact on computational acoustics, electromagnetics, and fluid‑structure interaction problems where cut‑cell meshes are attractive.