Stability analysis of very high order minimization-based and Taylor-based embedded boundary treatments of discontinuous Galerkin for hyperbolic equations

TL;DR

The paper analyzes stability of very high order embedded boundary treatments for DG discretizations of the 1D linear advection equation. It reframes the ROD minimization-based boundary treatment as a polynomial correction and compares it with the SB Taylor-based correction, examining explicit and implicit time integration across polynomials up to degree 6. A unified framework is developed to express ROD variants as polynomial corrections, enabling rigorous eigenvalue-based stability analysis and CFL assessment, with numerical experiments validating the theoretical predictions. The findings show that ROD-L^2 generally offers superior stability at high order, implicit time stepping enhances stability, but very high orders still face distance- and CFL-dependent constraints, especially for internal boundaries (d>0).

Abstract

In this paper, we present a stability analysis of very high order embedded boundary methods, specifically the Reconstruction for Off-site Data (ROD) and Shifted Boundary (SB) methods, coupled with a discontinuous Galerkin discretization for the linear advection equation. In unfitted configurations, these methods impose consistent modified boundary conditions on the computational boundary. Due to the high algebraic complexity of very high order schemes, the stability is studied by visualizing the eigenspectrum of the discretized operators. A recent study on the SB method demonstrated that its Taylor expansion can be formulated as a direct polynomial correction. In this work, we prove that the ROD minimization problem admits an analogous polynomial correction. This unified perspective provides significant benefits: algorithmically, it greatly simplifies the implementation of ROD by eliminating the need for linear system inversions at each iteration; mathematically, it enables a rigorous stability study. For completeness, a side-by-side stability analysis of the ROD and SB methods is presented for polynomials up to degree 6. Furthermore, due to the stability restrictions of embedded methods for hyperbolic problems, a coupling with both explicit and implicit time integration is investigated. A set of numerical experiments confirms the findings of the stability study.

Figures (12)

• Figure 1: Mesh configuration: tessellation of one-dimensional elements with a boundary condition (red circle) imposed at a distance $d$ from the left-most interface of the domain, which is considered as the computational boundary of the problem.
• Figure 2: Boundary position: the real boundary position can be either outside the first mesh element (above), or inside the first mesh element (below). In the first case, the distance will be taken as negative $d<0$, while for the second one the distance is taken as positive $d>0$.
• Figure 3: Amplification factor: periodic boundary conditions for the explicit discontinuous Galerkin method. When the amplification factor becomes greater than 1, the scheme is unstable.
• Figure 4: Stability region: SB polynomial correction of the homogeneous Dirichlet condition for the explicit discontinuous Galerkin method. Green areas are stable, while red areas are unstable.
• Figure 5: Stability region: ROD-E correction of the homogeneous Dirichlet condition for the explicit discontinuous Galerkin method. Green areas are stable, while red areas are unstable.

Theorems & Definitions (12)

• Proposition 4.1
• proof
• Remark 4.1: Efficiency of ROD polynomial correction
• Remark 4.2: Similarities with the SB method
• Remark 4.3: Consistency with body fitted boundary conditions
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Remark 4.4