Simultaneous energy and entropy stability of volume-penalisation immersed-boundary methods for the compressible Navier--Stokes equations
Abstract
Immersed-boundary methods avoid body-fitted meshes, but the energy or entropy stability of the underlying flow discretisation does not automatically extend to the immersed treatment. This manuscript analyses the volume-penalised compressible Navier--Stokes equations for both energy and entropy stability, on the continuous and semi-discrete levels, and integrates the isolated source exactly in time. The source is formulated as a one-parameter family indexed by θ, whose momentum component relaxes the fluid velocity towards the prescribed solid velocity; θ sets how much of the resulting kinetic-energy loss leaves the domain rather than being converted to internal energy. The two stability requirements impose opposing bounds on θ, with dissipation of the relative total energy requiring θ≥0 and stability of the mathematical entropy requiring θ≤1. Where penalisation acts, both conditions are therefore satisfied if and only if 0≤θ≤1, and the symmetric value θ=1/2 divides the kinetic-energy loss equally between the domain and the internal energy. We conclude that any entropy-stable split-form discontinuous Galerkin spectral element method remains semi-discretely entropy stable once the source is added at the collocation nodes, for stationary and prescribed moving bodies alike.
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BibTeX
@misc{ballout2026simultaneous,
title = {{Simultaneous energy and entropy stability of volume-penalisation immersed-boundary methods for the compressible Navier--Stokes equations}},
author = {Abbas Ballout and Albert Jiménez-Ramos and Gonzalo Rubio and Esteban Ferrer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17973},
eprint = {2610.17973},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17973},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Simultaneous energy and entropy stability of volume-penalisation immersed-boundary methods for the compressible Navier--Stokes equations
AU - Abbas Ballout
AU - Albert Jiménez-Ramos
AU - Gonzalo Rubio
AU - Esteban Ferrer
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17973
UR - https://arxiv.org/abs/2610.17973
AB - Immersed-boundary methods avoid body-fitted meshes, but the energy or entropy stability of the underlying flow discretisation does not automatically extend to the immersed treatment. This manuscript analyses the volume-penalised compressible Navier--Stokes equations for both energy and entropy stability, on the continuous and semi-discrete levels, and integrates the isolated source exactly in time. The source is formulated as a one-parameter family indexed by θ, whose momentum component relaxes the fluid velocity towards the prescribed solid velocity; θ sets how much of the resulting kinetic-energy loss leaves the domain rather than being converted to internal energy. The two stability requirements impose opposing bounds on θ, with dissipation of the relative total energy requiring θ≥0 and stability of the mathematical entropy requiring θ≤1. Where penalisation acts, both conditions are therefore satisfied if and only if 0≤θ≤1, and the symmetric value θ=1/2 divides the kinetic-energy loss equally between the domain and the internal energy. We conclude that any entropy-stable split-form discontinuous Galerkin spectral element method remains semi-discretely entropy stable once the source is added at the collocation nodes, for stationary and prescribed moving bodies alike.
ER -