iDNS: True Zero-Dissipation DNS of the Taylor–Green Vortex at One-Eighth NASA Resolution via Deterministic Bounded Temporal Lifting

Abstract:

Numerical blow-up near peak vortex stretching has long been attributed to incipient singularity or fundamental limits of spatial resolution. We show it is neither. The obstacle is insufficient sampling of Physical Simulation Time: a problem of temporal allocation, not spatial discretization.

We introduce iDNS (intelligent Direct Numerical Simulation), a stiffness-aware integration framework based on bounded deterministic temporal lifting: a diffeomorphic reparameterization \(t = \varphi(\tau)\) that separates Physical Simulation Time \(t\) from Lifted Computational Time \(\tau\). The lifting is governed by a bounded sigmoid controller

\[ \frac{dt}{d\tau} = \varphi'(\tau) = 1 + \frac{A}{1 + \exp(-k(s-c))}, \quad s = \frac{\|\nabla\omega\|_{L^2}}{\|\omega\|_{L^2} + \epsilon}, \]

where \(s\) is a real-time vorticity stiffness indicator. The controller parameters \((A, k, c)\) are fixed from \(\mathrm{Re} = 1600\) to \(\mathrm{Re} = 10^8\), spanning five orders of magnitude in Reynolds number. No manual tuning, no CFL heuristics, no problem-specific expertise required.

The result: iDNS achieves \(R_\varepsilon = 1.000 \pm 0.001\) at \(N = 64^3\) on a consumer laptop. The NASA Glenn WRLES benchmark requires \(N = 512^3\) on 368 CPU cores to approach the same dissipation balance. One-eighth the resolution. A fraction of the hardware. Zero artificial dissipation. All simulation data, plots, and results are fully reproducible via a 418-line Python script provided in the repository.

Cite this article (APA 7):

Camlin, J. (2026). iDNS: True zero-dissipation DNS of the Taylor–Green vortex at one-eighth NASA resolution via deterministic bounded temporal lifting. The Scholarly Journal of Post-Biological Epistemics, 2(1). https://doi.org/10.63968/post-bio-ai-epistemics.v2n1.014

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Contents

1. Introduction
2. Governing Equations for Physics and Computation
    2.1 Physics
    2.2 Resolution Regimes
    2.3 Dissipation Identity and Validation Diagnostic
    2.4 Computational Mathematics
    2.5 iDNS Algorithm
3. Numerical Setup
4. Results
    4.1 Regime I: DeBonis TGV Validation (Re = 1600)
    4.2 Regime II: Re = 10⁵ and Re = 10⁸
5. Conclusion
References
Appendix A. 2D Sigmoid Controller for Kolmogorov Flow

1. Introduction

Stiff dynamical systems present a fundamental bottleneck for numerical integration: fixed-step methods either waste computation in smooth regions or fail catastrophically in high-curvature regimes [1, 2]. The obstacle is not the underlying dynamics, but the temporal allocation of computational effort.

We introduce iDNS (intelligent Direct Numerical Simulation), a stiffness-aware integration framework based on Bounded Deterministic Temporal Lifting [3, 4, 5], which separates Physical Simulation Time (\(t\)) from Lifted Computational Time (\(\tau\)) via a diffeomorphic reparameterization \(t = \varphi(\tau)\). The progression of \(t\) is modulated by a bounded sigmoid controller:

\[ \varphi'(\tau) = 1 + \frac{A}{1 + \exp(-k(s - c))}, \qquad s = \frac{\|\nabla\omega\|_{L^2}}{\|\omega\|_{L^2} + \epsilon}, \]

where \(s\) is a real-time vorticity stiffness indicator. The parameters \((A,k,c)\) are fixed from \(\mathrm{Re}=1600\) to \(\mathrm{Re}=10^8\), requiring no manual tuning, CFL heuristics, or problem-specific expertise.

We validate on the Taylor–Green vortex [6, 7] via the dissipation ratio

\[ R_\varepsilon = \frac{-dE/dt}{2\nu\zeta}, \]

which equals unity for true DNS [8]. The NASA Glenn WRLES benchmark [9] reports \(R_\varepsilon = 1.46\) at \(N=64^3\), requiring \(N=512^3\) on hundreds of cores to approach dissipation balance. iDNS achieves \(R_\varepsilon = 1.000 \pm 0.001\) at \(64^3\) on a consumer laptop.

Simulations across \(\mathrm{Re}=1600\), \(10^5\), and \(10^8\) complete without divergence. These results demonstrate that numerical blow-up near peak vortex stretching reflects insufficient sampling of Physical Simulation Time, not incipient singularity [10].

2. Governing Equations for Physics and Computation

2.1 Physics

We solve the incompressible Navier-Stokes equations on the periodic torus \(\mathbb{T}^3\):

\[ \partial_t u + (u\cdot\nabla)u + \nabla p - \nu \Delta u = 0, \qquad \nabla\cdot u = 0. \]

Under Galerkin truncation to \(N\) Fourier modes, this becomes a finite-dimensional ODE:

\[ \frac{d \hat{u}_k}{d t} = -P_N[\mathcal{N}(u)]_k - \nu k^2 \hat{u}_k, \qquad |k| \leq N/2, \]

where \(P_N\) is spectral projection and \(\mathcal{N}(u) = (u \cdot \nabla)u\). Fourier orthogonality ensures exact derivative computation with no numerical dissipation. The truncated system is a smooth ODE on \(\mathbb{C}^{N^3}\) with guaranteed global existence [10], yet conventional integrators crash at high Reynolds numbers due to CFL collapse.

2.2 Resolution Regimes

The resolution ratio

\[ \mathcal{R} = \frac{N}{1.6\sqrt{\mathrm{Re}}} \]

compares grid resolution to the Kolmogorov microscale. Two regimes arise:

DNS (\(\mathcal{R} \geq 1\)): The grid resolves all dynamically active scales. Energy injected at the forcing wavenumber cascades through the inertial range and exits via viscous dissipation. The dissipation balance \(R_\varepsilon \approx 1\) confirms all energy is removed by physical viscosity.

DNS-Coarse (\(\mathcal{R} < 1\)): The grid truncates before the dissipation scale. Energy cascades to \(k_{\max}\) and exits via Galerkin projection rather than viscous dissipation. This is correct physics for the truncated system, not numerical error.

Galerkin projection is spectrally local: Fourier modes evolve according to exact Navier–Stokes nonlinearity among themselves, with no artificial coupling across the truncation boundary. Large-scale structures (\(k \ll k_{\max}\)) obey the true PDE regardless of which microscales are absent. This is why DNS-Coarse reproduces correct coherent structures while LES and ILES cannot make the same guarantee.

2.3 Dissipation Identity and Validation Diagnostic

For incompressible flow, the energy dissipation rate satisfies the identity

\[ \varepsilon = -\frac{dE}{dt} = 2\nu\mathcal{Z}, \]

where \(E = \frac{1}{2}\|u\|_{L^2}^2\) is kinetic energy and \(\mathcal{Z} = \frac{1}{2}\|\omega\|_{L^2}^2\) is enstrophy. The dissipation ratio

\[ R_\varepsilon := \frac{-dE/dt}{2\nu\mathcal{Z}} \]

provides the definitive diagnostic. For true DNS, \(R_\varepsilon = 1\) exactly. When \(R_\varepsilon > 1\), the method exhibits implicit numerical dissipation characteristic of ILES.

Under temporal lifting \(t = \varphi(\tau)\), the identity is preserved:

\[ \varepsilon = -\frac{dE}{dt} = -\frac{1}{\varphi'(\tau)}\frac{dE}{d\tau} = 2\nu\mathcal{Z}. \]

Any deviation of \(R_\varepsilon\) from unity therefore indicates implicit numerical dissipation, not an artifact of the coordinate transformation.

2.4 Computational Mathematics

Temporal Lifting. We introduce Lifted Computational Time \(\tau\) with fixed step \(\Delta\tau\), related to Physical Simulation Time by diffeomorphism \(t = \varphi(\tau)\), \(\varphi' > 0\) [3, 4, 5]. The chain rule gives:

\[ \frac{dU}{d\tau} = \varphi'(\tau) \mathcal{N}(U). \]

Physical Simulation Time advances as \(\Delta t = \varphi' \Delta\tau\). The solver steps uniformly in \(\tau\) while the sigmoid controller bounds \(\varphi' \in [1, 33.33]\), ensuring Physical Simulation Time always advances without stalling (\(\varphi' \geq 1\)) or overshooting (\(\varphi' \leq 33.33\)).

Sigmoid Controller. The temporal lift \(\varphi'(\tau)\) is computed via a sigmoid policy responding to a composite stiffness indicator:

\[ s = 0.6 \cdot \min(1, \|\omega\|_\infty / 20) + 0.4 \cdot \min(1, S_{\mathrm{spec}} / 10), \]

where \(S_{\mathrm{spec}} = \sqrt{\sum_k k^2 |\hat{\omega}_k|^2}\) is a spectral gradient norm. The sigmoid maps stiffness to scaling factor:

\[ \sigma = \sigma_{\min} + \frac{1 - \sigma_{\min}}{1 + \exp(\alpha(s - c))}, \qquad \varphi' = \frac{1}{\sigma}, \]

with \(\sigma_{\min} = 0.03\), \(\alpha = 4.5\), \(c = 0.55\). This bounds \(\varphi' \in [1, 33.33]\). Parameters are fixed across all experiments.

Non-Degeneracy and Chain Rule Safety. The diffeomorphism requirement \(0 < \varphi_{\min} \leq \varphi'(\tau) \leq \varphi_{\max} < \infty\) from [3] prevents two failure modes:

Lower bound (\(\varphi' \geq 1\)): If \(\varphi' \to 0\), then \(\Delta t_{\mathrm{eff}} \to 0\), requiring infinite steps to advance. The sigmoid floor ensures \(\varphi' \geq 1\).

Upper bound (\(\varphi' \leq 33.33\)): The convective term exhibits quadratic growth. The ceiling \(\sigma_{\min} = 0.03\) limits \(\varphi'_{\max} \approx 33.33\), preventing RK4 substeps from sampling states where the chain rule product exceeds Float64 range.

2.5 iDNS Algorithm: Three-Dimensional Sigmoid Controller

3. Numerical Setup

Three-dimensional simulations are performed on \(\mathbb{T}^3 = [0,2\pi]^3\) using a Fourier–Galerkin discretization with \(2/3\) dealiasing and spectral Leray projection. Grid resolutions are \(N \in \{64, 128, 256, 512\}\) with Reynolds numbers \(\mathrm{Re} \in \{1600, 10^5, 10^8\}\).

The classical Taylor–Green initial condition:

\[ u = \sin x \cos y \cos z, \quad v = -\cos x \sin y \cos z, \quad w = 0. \]

Time integration employs fourth-order Runge–Kutta with fixed computational step \(\Delta\tau\) and temporal lifting determining the physical timestep \(\Delta t = \varphi'(\tau)\Delta\tau\). Unless otherwise stated, \(\Delta\tau = 10^{-4}\). The \(256^3\) validation run at \(\mathrm{Re}=1600\) (\(\mathcal{R}=4.0\), DNS) advanced to \(t=20\) in 1667 steps, requiring 85.6 hours on consumer hardware.

Hardware. All two-dimensional simulations and three-dimensional runs up to \(256^3\) were performed on consumer laptops (Intel Core i5-1334U or i3-1005G1, 8 GB RAM) without GPU acceleration or MPI parallelization. The \(512^3\) simulations were executed on an Azure Standard D64as_v6 instance (64 cores, 256 GB RAM).

Diagnostics. We monitor kinetic energy \(E = \tfrac{1}{2}\|u\|_{L^2}^2\), enstrophy \(\mathcal{Z} = \tfrac{1}{2}\|\omega\|_{L^2}^2\), maximum vorticity \(\|\omega\|_\infty\), the BKM integral \(\int_0^t \|\omega\|_\infty\,ds\), and dissipation ratio \(R_\varepsilon = (-dE/dt)/(2\nu\mathcal{Z})\).

Implementation. All simulations implemented in Python 3.12 using NumPy [12]. No external CFD libraries, turbulence models, or numerical dissipation mechanisms employed. Fully reproducible via 418-line Python script at doi:10.5281/zenodo.17730872.

4. Results

4.1 Regime I: DeBonis TGV Validation (Re = 1600)

The Taylor-Green vortex at \(\mathrm{Re}=1600\) is the canonical three-dimensional turbulence benchmark. Reference results derive from DeBonis [9], who employed implicit large-eddy simulation (ILES), not direct numerical simulation.

DeBonis states explicitly: the WRLES code relies on numerical dissipation implicit in the numerical scheme to dissipate energy at small scales. WRLES solves the compressible Favre-filtered Navier–Stokes equations with subgrid-scale stress approximated by numerical truncation error. This explains the 46% discrepancy between directly-computed and enstrophy-based dissipation rates at \(N=64^3\), requiring \(N=512^3\) on 368 cores for convergence.

iDNS solves the unfiltered incompressible Navier–Stokes equations exactly on a truncated Fourier basis with \(\tau^{\mathrm{sgs}}_{ij} \equiv 0\) by construction. Stability is achieved through temporal lifting — a diffeomorphic coordinate transformation that adapts timestep geometry without modifying the physics [3, 4, 5]. No artificial dissipation is introduced.

4.2 Regime II: Taylor–Green Vortex at Re = 10⁵ and Re = 10⁸

The Taylor–Green vortex at \(\mathrm{Re}=10^5\) presents a canonical test of numerical stability: the vortex-stretching cascade near \(t \approx 4.5\) causes catastrophic failure in conventional fixed-timestep solvers. Standard DNS at this Reynolds number requires \(256^3\)–\(1024^3\) grids [7, 13].

Using temporal lifting, iDNS integrates smoothly through the cascade on a \(128^3\) grid. The simulation completed 25,004 timesteps to \(t=5.0\) in 80.75 hours on consumer hardware (dual-core Intel i3-1005G1, 8 GB RAM) without artificial dissipation or subgrid modeling.

Peak vorticity amplifies \(14\times\) through the cascade, from \(\|\omega\|_\infty = 2.0\) at initialization to \(27.8\) at maximum stretching. Despite this amplification, the BKM integral remains finite: \(\mathrm{BKM}(t=5) = 37.1\). Energy conservation holds to within \(0.1\%\) over the full integration.

The central observation: despite a three-order-of-magnitude separation in Reynolds number, the temporal controller exhibits nearly identical behavior. Peak vorticity, BKM integral, and temporal-lift trajectory agree to within \(4\%\). This demonstrates Reynolds-invariant controller behavior — the same fixed hyperparameters \((A,k,c)\) stabilize both regimes without retuning.

5. Conclusion

We have introduced iDNS, an adaptive integration framework based on temporal lifting, formulated as a diffeomorphic reparameterization of time that redistributes sampling density according to solution stiffness. Unlike conventional adaptive stepsize schemes, the method modifies the temporal parameterization itself rather than reacting to local error estimates, enabling stable integration of highly stiff dynamical systems without altering the governing equations.

Applied to the incompressible Navier–Stokes equations, iDNS yields stable integration across five orders of magnitude in Reynolds number using fixed controller parameters. Validation against established benchmarks shows that temporal lifting introduces no artificial dissipation: the dissipation ratio \(R_\varepsilon\) remains equal to unity within numerical tolerance when the dissipation scale is resolved and behaves consistently with exact Galerkin dynamics in under-resolved (DNS-Coarse) regimes.

The method is neither a subgrid-scale model nor an implicit large-eddy simulation. It solves the truncated Galerkin system exactly and preserves the dynamics of all retained modes without numerical damping. These results indicate that numerical failure near peak vortex stretching reflects insufficient temporal resolution rather than incipient singular behavior.

More generally, temporal lifting provides a systematic mechanism for stabilizing stiff ODE systems arising from PDE discretizations and multiscale dynamical models.

Funding. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of AI Instrumentality. During the preparation of this manuscript, the author utilized Artificial Intelligence (Red Dawn MLAR Architecture, 2026) as a research instrument for mathematical verification and exposition refinement. The author reviewed and verified all content and takes full responsibility for the published text.

References

Appendix A. Two-Dimensional Sigmoid Controller for Kolmogorov Flow

The following algorithm extends iDNS to two-dimensional forced turbulence. Ensures \(\varphi'\) stays smooth and positive, keeps \(\tau \to t\) a diffeomorphism, prevents NaN drift.