Minimization-based embedded boundary methods as polynomial corrections: a stability study of discontinuous Galerkin for hyperbolic equations

TL;DR

This paper introduces a unified theoretical framework for high order embedded boundary methods, simplifying their implementation and establishing the first stability results for the ROD method with DG discretization of hyperbolic equations.

Contribution

It reveals that ROD minimization can be reformulated as polynomial correction, leading to computational simplification and the first stability analysis for ROD with DG methods.

Findings

Reformulation of ROD as polynomial correction simplifies implementation.

First stability proof for ROD with DG for hyperbolic equations.

Eigenvalue analysis characterizes stability regions up to polynomial degree six.

Abstract

This work establishes a novel, unified theoretical framework for a class of high order embedded boundary methods, revealing that the Reconstruction for Off-site Data (ROD) treatment shares a fundamental structure with the recently developed shifted boundary polynomial correction [Ciallella, M., et al. (2023)]. By proving that the ROD minimization problem admits an equivalent direct polynomial correction formulation, we unlock two major advances. First, we derive a significant algorithmic simplification, replacing the solution of the minimization problem with a straightforward polynomial evaluation, thereby enhancing computational efficiency. Second, and most critically, this reformulation enables the first stability result for the ROD method when applied to the linear advection equation with discontinuous Galerkin discretization. Our analysis, supported by a comprehensive eigenspectrum study for polynomial degrees up to six, characterizes the stability region of the new ROD formulation. The theoretical findings, which demonstrate the stability constraints, are validated through targeted numerical experiments.