An Energy-based discontinuous Galerkin method for the Westervelt equation with absorbing boundary conditions
Abstract
This paper develops and analyzes an energy-based discontinuous Galerkin method for the Westervelt equation in the acoustic-velocity-potential formulation with spatially varying material parameters and nonlinear absorbing boundary conditions. The semi-discrete scheme is constructed from the underlying energy balance and incorporates an additional consistent term to accommodate the strong damping operator within the discrete energy framework. Numerical fluxes are designed at both interelement and physical boundaries to provide a unified treatment of interior and boundary dissipation. For the nonlinear analysis, we study an auxiliary variable-coefficient linearized problem and establish stability and a priori error estimates, including the additional terms induced by the solution-dependent absorbing boundary flux. These estimates are combined with a fixed-point argument to prove existence, uniqueness, and convergence of the nonlinear semi-discrete approximation under suitable regularity, smallness, and nondegeneracy assumptions. Numerical experiments confirm the theoretical convergence results and illustrate the performance of the proposed method.
Related
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BibTeX
@misc{wang2026energybased,
title = {{An Energy-based discontinuous Galerkin method for the Westervelt equation with absorbing boundary conditions}},
author = {Yiwei Wang and Lu Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38058},
eprint = {2610.38058},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38058},
}
RIS
TY - UNPB
TI - An Energy-based discontinuous Galerkin method for the Westervelt equation with absorbing boundary conditions
AU - Yiwei Wang
AU - Lu Zhang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.38058
UR - https://arxiv.org/abs/2610.38058
AB - This paper develops and analyzes an energy-based discontinuous Galerkin method for the Westervelt equation in the acoustic-velocity-potential formulation with spatially varying material parameters and nonlinear absorbing boundary conditions. The semi-discrete scheme is constructed from the underlying energy balance and incorporates an additional consistent term to accommodate the strong damping operator within the discrete energy framework. Numerical fluxes are designed at both interelement and physical boundaries to provide a unified treatment of interior and boundary dissipation. For the nonlinear analysis, we study an auxiliary variable-coefficient linearized problem and establish stability and a priori error estimates, including the additional terms induced by the solution-dependent absorbing boundary flux. These estimates are combined with a fixed-point argument to prove existence, uniqueness, and convergence of the nonlinear semi-discrete approximation under suitable regularity, smallness, and nondegeneracy assumptions. Numerical experiments confirm the theoretical convergence results and illustrate the performance of the proposed method.
ER -