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A Reynolds-Semi-Robust, Globally Divergence-Free E-HDG/IMEX-SAV Method for Variational Initial-State Data Assimilation of the Navier-Stokes Equations

Ya Min, Xian Zhang, Xiaoping Xie

arXiv:2609.17096v1math.NAcs.NA

TL;DR

Variational initial-state assimilation for nonlinear Navier-Stokes flows lacks rigorous fully discrete analysis that jointly addresses high-Reynolds-number regimes, exact incompressibility, and spatial and temporal errors. The paper develops an OTD E-HDG/IMEX-SAV method and proves conditional local solvability and Reynolds-semi-robust velocity error bounds, while the forward state scheme is unconditionally energy stable.

  • Problem

    Rigorous fully discrete analysis remains limited for nonlinear Navier-Stokes initial-state assimilation that jointly addresses high-Reynolds-number regimes, exact discrete incompressibility, and spatial and temporal errors.

    Source: §1 · pp. 1–2

  • Method

    The paper combines an optimize-then-discretize formulation with an exactly divergence-free E-HDG spatial method and a first-order IMEX-SAV time integrator.

    Source: §1 · p. 3, §6 · p. 48

  • Results

    Under the stated assumptions and conditions, the discrete optimality system has a locally unique solution and the initial, state, and adjoint velocity L2 errors are O(h^k + Δt), with constants containing no explicit negative powers of viscosity.

    Source: §6 · p. 48

  • Takeaways & Limitations

    The forward state scheme is unconditionally energy stable, and experiments report convergence, machine-precision discrete incompressibility, effective nonlinear-solver performance, and stable behavior in small-viscosity tests.

    Source: §1 · p. 3, §6 · p. 48

    The local-solvability proof uses a sufficient Tikhonov-parameter contractivity condition, which the paper does not claim is necessary.

    Source: Step 3: Local solvability and error… · p. 37

Abstract

from arXiv · show

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 $\Delta t\lesssim h^2$, 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 $O(h^k+\Delta t)$ 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.

1 Introduction

The paper addresses the limited rigorous analysis of fully discrete variational initial-state assimilation for nonlinear Navier–Stokes flows, especially at high Reynolds numbers. It develops an OTD E-HDG/IMEX-SAV method and proves conditional Reynolds-semi-robust error bounds alongside unconditional energy stability of the forward scheme.

1 Introduction

  • The work targets a gap in rigorous fully discrete analysis of nonlinear Navier-Stokes initial-state assimilation that treats high-Reynolds-number regimes, exact discrete incompressibility, and spatial and temporal errors together.The introduction also notes that existing analytical studies have mainly addressed linearized incompressible models.Source: §1 · p. 2
  • The E-HDG design uses viscosity-independent facet stabilization with velocity-trace jumps instead of upwind convection stabilization to control convection-related terms without negative viscosity powers.The method also uses a continuous velocity trace and discontinuous pressure trace to retain globally divergence-free velocities while reducing globally coupled degrees of freedom.Source: §1 · p. 3
  • The method combines OTD with E-HDG spatial discretization and IMEX-SAV time stepping for variational initial-state assimilation of the incompressible Navier-Stokes equations.The E-HDG spaces yield globally divergence-free state and adjoint velocities and a divergence-free discrete initial velocity.Source: §1 · p. 3
  • Under regularity and local-trajectory assumptions, a mesh-dependent time-step condition, and sufficient Tikhonov-parameter contractivity, the discrete OTD solution exists locally and uniquely, with L2 velocity errors of O(h^k + Δt).The bounds cover the initial, state, and adjoint velocities, and their constants contain no explicit negative powers of viscosity.Source: §1 · p. 3
  • The analysis addresses the nonlinear coupling by separating discretization error along a reference trajectory from perturbation error caused by changing the discrete initial value.The initial condition is the control variable, and its errors propagate through the state trajectory into the adjoint system.Source: §1 · pp. 2–3
  • The forward IMEX-SAV state scheme is unconditionally energy stable, while the time-step condition in the error analysis controls nonlinear error terms.The analytical time-step restriction does not arise from the forward scheme’s energy stability.Source: §1 · pp. 3–4

2 Data assimilation problem

This section sets the notation and formulates the continuous initial-state assimilation problem, its adjoint and optimality conditions, and an equivalent SAV state representation. It then gives the corresponding weak formulations.

2.1 Notation

  • The notation includes Sobolev and Bochner spaces, the solenoidal L2 space of divergence-free fields with vanishing normal trace, and generic constants independent of mesh size, time step, and viscosity unless stated otherwise.For D = Ω, the domain subscript is omitted from Sobolev norms and seminorms.Source: §2.1 · pp. 4–5

2.2 Continuous adjoint system and optimality conditions

  • The continuous problem treats the initial velocity as the control, defines the initial-to-state map and reduced functional, and derives an adjoint system and first-order optimality condition.The Tikhonov term is coercive, and the paper states that the continuous problem has at least one global minimizer.Source: §2.2 · pp. 5–6

2.3 Equivalent SAV reformulation of the state system

  • The SAV reformulation introduces a scalar auxiliary variable to construct the IMEX-SAV time discretization, but that variable is determined at the continuous level and does not add an independent optimization variable or adjoint.The reformulated state system is stated to be equivalent to the original Navier-Stokes system at the continuous level.Source: §2.3 · p. 6

2.4 Weak formulations

  • The weak formulations state the forward state equation, backward adjoint equation, and first-order optimality condition; the two adjoint convection terms come from linearizing the state convection term.The reduced optimality residual is introduced so stationary points correspond to its zeros.Source: §2.4 · p. 7

3 Fully discrete E-HDG method

This section presents the fully discrete E-HDG/IMEX-SAV formulation and analyzes the resulting discrete state and adjoint equations. It covers divergence-free velocities and state energy stability, as well as adjoint well-posedness and stability, with adjoint control depending on state-gradient bounds.

3.1 Mesh-related notation and discrete spaces

  • The E-HDG spaces use a globally continuous facet velocity trace and a discontinuous facet pressure trace, reducing globally coupled velocity-trace degrees of freedom relative to standard HDG.The element velocity and pressure spaces have polynomial degrees k and k−1, respectively.Source: §3.1 · p. 8

3.2 Fully discrete IMEX-SAV E-HDG scheme

  • The discrete state and adjoint velocities are exactly divergence-free and belong to H0(div,Ω), and the reconstructed initial velocity is also exactly divergence-free.These properties follow from elementwise incompressibility and single-valued normal traces across interior facets.Source: §3.2 · pp. 10–11

3.3 Consistency

  • The discrete forms are spatially consistent with the continuous Navier–Stokes equations; divergence-free flow and trace continuity cancel the convective boundary contribution.The consistency identity is stated for all discrete velocity and pressure test functions.Source: §3.3 · pp. 11–12

3.4 Stability properties of the discrete forms and subproblems

  • The state equation has a unique solution at every time level and is unconditionally energy stable, with an energy bound depending on the initial value and a constant independent of h, Δt, and ν.The stability result applies for any given discrete initial value.Source: §3.4 · p. 13
  • The discrete adjoint problem has a unique solution at every backward time level and satisfies a stability estimate.The estimate is Reynolds-semi-robust when the coefficient bound is independent of h and Δt and contains no explicit negative powers of ν; otherwise its constant may contain negative powers of ν.Source: §3.4 · p. 15
  • Adjoint stability and error estimates involve state-gradient norms because the linearized Navier–Stokes operator contributes a term involving the state gradient.The corresponding state-gradient bounds are treated as regularity assumptions and are recorded explicitly in the subsequent local error analysis.Source: §3.4 · p. 17

4 Existence, uniqueness, and error analysis

The analysis establishes local existence and uniqueness of the discrete OTD system using a Banach fixed-point argument, beginning with a reference-trajectory error estimate under regularity, local-trajectory, and time-step assumptions. It also proves Reynolds-semi-robust error bounds of order O(h^k + Δt) for the reconstructed initial velocity, state, and adjoint, with no explicit negative powers of viscosity.

4 Existence, uniqueness, and error analysis

  • The analysis assumes regularity of the reference trajectory, uniform local bounds for discrete state and adjoint trajectories, and a mesh-dependent time-step restriction.The local bounds introduce no explicit negative powers of viscosity.Source: §4 · pp. 17–18

4.1 Preliminary material

  • Preliminary tools include Raviart–Thomas projection properties preserving discrete divergence, initial-facet reconstruction, and a convection-consistency estimate.These estimates support the error analysis for the discretization.Source: §4.1 · pp. 19–20
  • The preliminary analysis also provides error estimates for the continuous and discrete state solutions under the stated regularity assumption.The supplied passage introduces these estimates but does not expose their formulas.Source: §4.1 · p. 21

4.2 Analysis

  • Local existence and uniqueness of the discrete OTD system follow from a Banach fixed-point argument combining a reference-trajectory residual estimate, local Lipschitz bounds, and contraction.The proof is organized in these three steps.Source: §4.2 · p. 22

Step 1: Reference-trajectory residual estimate

  • The first proof step derives a reference-state error identity and estimates the IMEX-SAV convection residual as part of bounding that trajectory error.The reference discrete initial value is obtained from a divergence-preserving Raviart–Thomas projection.Source: Step 1: Reference-trajectory residual… · pp. 23–24
  • Under Assumptions 4.1–4.2 and the time-step condition, the reference discrete state satisfies an error estimate with a constant independent of n and h and containing no negative powers of viscosity.The estimate is stated for sufficiently small h and Δt.Source: Step 1: Reference-trajectory residual… · p. 26
  • The reference-trajectory residual analysis yields state, adjoint, and reconstructed-initial velocity error estimates with constants independent of h and Δt and no explicit negative powers of viscosity.The estimates rely on Assumptions 4.1 and 4.2 and the time-step condition (4.6).Source: Step 1: Reference-trajectory residual… · pp. 29, 31

Step 2: Local Lipschitz continuity of the discrete state and adjoint maps

  • Within the local discrete neighborhood, the state and adjoint maps are locally Lipschitz with constants independent of h and Δt and containing no negative powers of viscosity.These bounds hold under Assumptions 4.1 and 4.2 and the time-step condition (4.6), for sufficiently small h and Δt.Source: Step 2: Local Lipschitz continuity of… · pp. 31, 34

Step 3: Local solvability and error estimates

  • The discrete OTD system has a unique local solution, with reconstructed-initial, state, and adjoint velocity errors of order O(h^k + Δt) and constants without explicit negative powers of viscosity.The result assumes the theorem’s small-mesh and time-step conditions and its contractivity hypothesis.Source: Step 3: Local solvability and error… · p. 36
  • The sufficient contractivity condition q_A = C_1 C_A/(2α) < 1 closes the fixed-point argument but is not claimed to be necessary for local solvability.A second-order sufficient optimality condition and discrete inverse-function analysis might weaken the restriction, but that analysis is beyond the paper’s scope.Source: Step 3: Local solvability and error… · p. 37

5 Numerical implementation and experiments

The experiments evaluate implementation efficiency, convergence, initial-velocity reconstruction, and cavity-flow behavior. They report high-order convergence and divergence-free reconstructions, with qualifications for boundary-layer regularity and nonhomogeneous cavity boundaries.

5.1 Implementation

  • Each IMEX-SAV time step can be implemented with two constant-coefficient Stokes-type solves followed by a scalar update.The scalar equation is the Schur-complement relation associated with the uniquely solvable state system.Source: §5.1 · p. 39
  • The L-BFGS-type update reaches efficiency comparable to the best-tuned fixed-point iteration and similar reconstruction accuracy without choosing a relaxation parameter.The fixed-point iteration’s convergence is sensitive to its relaxation parameter.Source: §5.1 · p. 39

5 Numerical implementation and experiments

  • The experiments use a uniform triangular mesh and include two exact-solution convergence tests plus oscillating-lid cavity data assimilation.Each square in the mesh is split along one diagonal, with h ≃ m^-1.Source: §5 · p. 37, §5.2 · p. 41

5.2 Numerical experiments

  • Observed rates approach k + 1 in low- and high-Reynolds regimes, one order above the proved O(h^k + Δt) bound; computed velocities remain divergence-free to machine precision.The convergence statement covers the two smooth manufactured-solution regimes reported in Tables 1 and 2.Source: Table 1 · p. 41, §5.2 · p. 41, Table 2 · p. 42
  • The boundary-layer test illustrates small-viscosity performance, not viscosity-uniform regularity verification; the nonhomogeneous cavity case lies outside the theoretical analysis.The layer-dependent Sobolev and W^1,∞ norms are not uniform in ν.Source: §5.2 · p. 43
  • At increasing Reynolds number, cavity flow develops more pronounced primary, secondary, corner, and counter-rotating vortices; computed patterns reproduce qualitative features reported in.The streamline phases include forward motion, post-reversal evolution, and maximum reverse motion.Source: §5.2 · p. 44
  • Initial-velocity error falls from 71.1% to 2.55% at ν = 10−2 and from 58.6% to 3.82% at ν = 10−3; divergence measures are of order 10−14.The reconstructed velocities also recover the reference initial fields’ principal vortex structures.Source: §5.2 · p. 44, Table 5 · p. 44, Figure 8 · p. 48

6 Conclusions

The fully discrete OTD method combines an exactly divergence-free E-HDG discretization with a first-order IMEX-SAV integrator. Under stated assumptions, it yields viscosity-semi-robust velocity error estimates and local existence and uniqueness for the discrete optimality system.

6 Conclusions

  • The E-HDG spaces yield globally divergence-free state and adjoint velocities and a divergence-free reconstructed initial velocity; the forward IMEX-SAV scheme is unconditionally energy stable.Static condensation leaves only trace unknowns in the globally coupled systems.Source: §6 · p. 48
  • Initial, state, and adjoint velocity L2 errors are O(h^k + Δt), with constants containing no explicit negative powers of viscosity.The estimates assume regularity and local-trajectory bounds that likewise introduce no explicit negative powers of viscosity.Source: §6 · p. 48
  • Under regularity and local-trajectory assumptions, Δt ≲ h^2, and a sufficient Tikhonov contractivity condition, the discrete OTD optimality system exists locally and uniquely.The contractivity condition comes from the fixed-point argument and is not necessary for the underlying continuous optimization problem.Source: §6 · p. 48
  • The analysis is local and relies on strong bounds for the reference and nearby discrete trajectories.The boundary-layer experiment is a stress test, not a viscosity-uniform verification of the analysis assumptions.Source: §6 · p. 48
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