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

TL;DR

This work develops an energy-preserving Domain of Dependence stabilization for cut-cell DG discretizations of the linear wave equation. By distinguishing central fluxes from dissipative surface terms and introducing propagation forms that extend the numerical domain of dependence, the method achieves energy conservation (or controlled dissipation) while enabling explicit time stepping based on the background mesh size. The authors prove energy stability/conservation for the stabilized scheme and demonstrate high accuracy and long-time stability through rotated-square and rotated-channel benchmarks, including careful treatment of reflecting boundaries. The framework paves the way for extensions to nonlinear systems and 3D problems, with potential applications to Euler-type hyperbolic balance laws and high-order cut-cell methods.

Abstract

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.

Figures (6)

• Figure 1: Characteristics entering $E_{\text{dw}}$ in the time interval $[t^n, t^{n+1}]$. Since the neighbor cell $E$ is too small, these characteristics even reach into $E_{\text{up}}$ at time $t^n$.
• Figure 2: Extension of the domain of dependence of a small cell $E$ via our stabilization method displayed for the linear transport equation with a transport vector $\beta$ which is parallel to the cut face. On the left we see the domain of dependence of the cell $E_{\text{dw}}$ when the base DG scheme \ref{['eq: base scheme']} is applied. On the right we see the domain of the dependence of the same element in the mesh when our stabilized scheme \ref{['eq: stabilized scheme']} is applied. Now the cell $E_{\text{dw}}$ will receive information from $E_{\text{up}}$ in a single time step/stage even when a large time step size is used.
• Figure 3: Test 1: Rotated square geometry and cut cell mesh example
• Figure 4: Numerical convergence results for polynomial degrees $1$, $2$ and $3$ on the rotated square. The three columns state the error in the pressure, first velocity and second velocity component, respectively. These errors are measured in both the $||\cdot||_{L^2}$ (first row) and $||\cdot||_{\infty}$ (second row) norm. In both cases the results where computed in double precision (FP64). While the convergence in the $||\cdot||_{L^2}$ norm shows the expected behavior, we see a reduced convergence rate and significant outliers in the $||\cdot||_{\infty}$. The last row again shows the error in the $||\cdot||_{\infty}$ but this time computed in extended double precision (FP80). Compared to the FP64 results, wee see a significant improvement, indicating that the outliers are caused by numerical precision issues.
• Figure 5: Rotated channel geometry and cut-cell mesh example

Theorems & Definitions (25)

• Definition 1: Mirroring operator
• Remark 1: Skew symmetry
• Lemma 1
• proof
• Definition 2: Cell classification
• Remark 2
• Definition 3: Extension operator
• Definition 4: Reflected extension operator
• Definition 5: Propagation forms
• Remark 3