A Reynolds-Semi-Robust, Globally Divergence-Free E-HDG/IMEX-SAV Method for Variational Initial-State Data Assimilation of the Navier-Stokes Equations
Abstract
This paper develops an embedded-hybridized discontinuous Galerkin (E-HDG) method combined with a first-order implicit-explicit scalar auxiliary variable (IMEX-SAV) time discretization for variational initial-state data assimilation governed by the unsteady incompressible Navier-Stokes equations. We adopt an optimize-then-discretize strategy. The spatial discretization uses discontinuous piecewise polynomials of degrees k and k−1 for the element velocity and pressure, respectively, a continuous degree-k velocity trace, and a discontinuous degree-k pressure trace. The resulting state and adjoint velocities, as well as the reconstructed initial velocity, are globally divergence-free. The forward IMEX-SAV state scheme is unconditionally energy stable. Under suitable regularity and local-trajectory assumptions whose bounds introduce no explicit negative powers of the viscosity, the mesh-dependent condition Δt≲ h2, and a contractivity condition on the Tikhonov parameter, we establish local existence and uniqueness of the fully discrete OTD optimality system and Reynolds-semi-robust L2 error estimates of order O(hk+Δt) for the state, adjoint and reconstructed initial velocities; the constants contain no explicit negative powers of the viscosity. Numerical experiments confirm convergence on smooth tests and demonstrate machine-precision discrete incompressibility, effective nonlinear-solver behavior, and stable performance in small-viscosity regimes.
Related
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BibTeX
@misc{min2026reynoldssemirobust,
title = {{A Reynolds-Semi-Robust, Globally Divergence-Free E-HDG/IMEX-SAV Method for Variational Initial-State Data Assimilation of the Navier-Stokes Equations}},
author = {Ya Min and Xian Zhang and Xiaoping Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17096},
eprint = {2609.17096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17096},
}
RIS
TY - UNPB
TI - A Reynolds-Semi-Robust, Globally Divergence-Free E-HDG/IMEX-SAV Method for Variational Initial-State Data Assimilation of the Navier-Stokes Equations
AU - Ya Min
AU - Xian Zhang
AU - Xiaoping Xie
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17096
UR - https://arxiv.org/abs/2609.17096
AB - This paper develops an embedded-hybridized discontinuous Galerkin (E-HDG) method combined with a first-order implicit-explicit scalar auxiliary variable (IMEX-SAV) time discretization for variational initial-state data assimilation governed by the unsteady incompressible Navier-Stokes equations. We adopt an optimize-then-discretize strategy. The spatial discretization uses discontinuous piecewise polynomials of degrees k and k−1 for the element velocity and pressure, respectively, a continuous degree-k velocity trace, and a discontinuous degree-k pressure trace. The resulting state and adjoint velocities, as well as the reconstructed initial velocity, are globally divergence-free. The forward IMEX-SAV state scheme is unconditionally energy stable. Under suitable regularity and local-trajectory assumptions whose bounds introduce no explicit negative powers of the viscosity, the mesh-dependent condition Δt≲ h^2, and a contractivity condition on the Tikhonov parameter, we establish local existence and uniqueness of the fully discrete OTD optimality system and Reynolds-semi-robust L^2 error estimates of order O(h^k+Δt) for the state, adjoint and reconstructed initial velocities; the constants contain no explicit negative powers of the viscosity. Numerical experiments confirm convergence on smooth tests and demonstrate machine-precision discrete incompressibility, effective nonlinear-solver behavior, and stable performance in small-viscosity regimes.
ER -