Clean Numerical Simulation (CNS)

• Clean Numerical Simulation (CNS) is a numerical method that uses high-order time integration and multiple-precision arithmetic to reliably simulate chaotic systems.
• It suppresses truncation and round-off errors well below physical uncertainty, ensuring convergent and reproducible trajectories over extended time intervals.
• CNS provides a rigorous framework for analyzing deterministic chaos and turbulence, offering reliable benchmarks and insights into error propagation in numerical simulations.

The Clean Numerical Simulation (CNS) is a numerical methodology designed to obtain mathematically reliable (i.e., convergent and reproducible) solutions of chaotic dynamical systems and spatiotemporal turbulence over finite, but arbitrarily long, time intervals. CNS achieves this by deliberately suppressing both truncation and round-off errors far below physically meaningful uncertainty—often below intrinsic fluctuations such as quantum or thermal noise—thereby generating clean trajectories whose deviation from the true solution is negligible for all practical purposes. Developed specifically to address the strong sensitive dependence on initial conditions (SDIC) characteristic of chaos, CNS underpins a new class of numerical experiment in rigorous chaos and turbulence research, permitting the study of both deterministic unpredictability and the propagation of micro-level physical uncertainty.

1. Motivation and Conceptual Rationale

The key motivation for CNS arises from the realization that in chaotic systems, any minute error—be it from discretization (truncation error) or finite precision (round-off error)—is exponentially amplified due to SDIC. In conventional numerical simulations (such as Direct Numerical Simulation, DNS) employing double-precision arithmetic and low-order integrators, the background numerical noise rapidly grows to pollute computed trajectories, ultimately dominating over any inherent micro-level physical uncertainty present in the initial data. This renders standard simulations unreliable for faithful prediction, analysis of statistical transients, or investigation of physical unpredictability (Li et al., 2016, Liao, 2013).

CNS is designed to reverse this failure mode by systematically reducing numerical noise below any prescribed threshold, thereby enabling clean simulation of the physical mechanisms through which micro-level uncertainties (e.g., Planck- or thermal-scale fluctuations) propagate into macroscopic randomness. This approach not only clarifies the boundary of computational predictability but also provides reliable benchmarks for DNS and other numerical methods (Liao, 2011, Li et al., 2016, Lin et al., 2016).

2. Mathematical Foundations and Error Suppression Strategy

At its core, CNS consists of three technical pillars:

a) High-Order Taylor Series Time Integration

A Taylor expansion of arbitrarily high order $M$ is employed to advance each ODE or PDE degree of freedom in time:

$$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$

Here, $u^{(k)}(t_n)$ denotes the $k$th time derivative, recursively generated via repeated differentiation of the differential equations. Increasing $M$ suppresses the local truncation error as $\mathcal{O}((\Delta t)^{M+1})$ (Li et al., 2016).

b) Multiple-Precision Arithmetic

All intermediate and final variables are represented using $N_s$ significant digits, often with $N_s \sim 60-1000$, depending on the problem's required accuracy and predictability horizon. This reduces round-off errors per operation from $\sim\!10^{-16}$ (double precision) to $\sim\!10^{-N_s}$ (Qin et al., 2023, Liao, 2011).

c) Explicit Error Control and Verification

The background numerical noise is defined as $$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$0, where $$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$1 is the stepwise truncation error and $$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$2 is the round-off error. Both are set to a user-specified level (typically much smaller than physical micro-uncertainty) via adaptive adjustment of $$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$3, $$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$4, and $$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$5. A posteriori convergence is verified by repeating the simulation with stricter parameters, ensuring all observables agree to within the chosen tolerance over $$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$6 (Li et al., 2016, Hu et al., 2019).

Table: Core Algorithmic Elements of CNS

Element	Technique	Typical Parameter Range
Time integration	Taylor series	$$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$7–$$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$8
Precision	Arbitrary-precision	$$u_{n+1} = u_n + \sum_{k=1}^{M} \frac{(\Delta t)^k}{k!} u^{(k)}(t_n) + \mathcal{O}((\Delta t)^{M+1}).$$9–$u^{(k)}(t_n)$0 digits
Error verification	Cross-simulation	Tolerance $u^{(k)}(t_n)$1 micro-fluctuation

3. Predictability Horizon and Noise Amplification

In chaotic dynamics, numerical noise is observed to grow as

$u^{(k)}(t_n)$2

where $u^{(k)}(t_n)$3 is a noise-growing exponent closely related to the largest Lyapunov exponent of the system. This exponential amplification defines a critical predictable time window

$u^{(k)}(t_n)$4

where $u^{(k)}(t_n)$5 is the maximum tolerable error (often set below physical noise). CNS ensures $u^{(k)}(t_n)$6 is so small that $u^{(k)}(t_n)$7 can be made as large as needed for rigorous simulations, typically extending the trustworthy window by orders of magnitude relative to DNS (Qin et al., 2023, Liao, 2013). For example, in the three-body problem, $u^{(k)}(t_n)$8 and $u^{(k)}(t_n)$9 digits yielded convergent trajectories out to $k$0, compared to classical divergence times $k$1 observed in double-precision integrators (Liao, 2013).

4. Benchmarking, Statistical Validity, and Physical Insight

CNS is particularly effective in producing reliable benchmarks to:

• Diagnose the influence of numerical noise on statistical observables (means, variances, PDFs, energy spectra) in both equilibrium and non-equilibrium turbulent flows.
• Disentangle true multiplicity and non-uniqueness phenomena from spurious artifacts of numerics (Liao et al., 13 Feb 2026).
• Examine the propagation and amplification of micro-level uncertainty into observable macroscopic randomness, with direct evidence for intrinsic unpredictability in Rayleigh–Bénard turbulence and the three-body problem (Lin et al., 2016, Liao, 2013).
• Provide rigorously converged trajectories and statistics for machine learning, where "clean" data are essential; ML trained on noisy data inherits biases not present in CNS-based training (Yang et al., 2021).

Notably, CNS has demonstrated that, for time-independent (stationary) statistics, numerical noise is often less consequential; however, for time-dependent (transient, non-equilibrium) statistics, uncontrolled numerical artifacts can qualitatively and quantitatively corrupt results—even for spatial resolutions and time-steps commonly regarded as "sufficient" by DNS criteria (Li et al., 2016, Qin et al., 2024, Qin et al., 15 Jul 2025).

5. Self-Adaptive and High-Dimensional CNS Algorithms

To accommodate computational demands in large-scale chaotic systems (high-dimensional PDEs), CNS has been extended with self-adaptive strategies (Qin et al., 2023):

• Dynamically balancing truncation and round-off errors at each stage, progressively relaxing overly tight tolerances as exponential error amplification renders initial strictness redundant.
• Automated time-step and Taylor order selection (e.g., via the Barrio variable-stepsize method), maintaining error thresholds while maximizing algorithmic efficiency.
• Staged reduction of multiple-precision arithmetic as $k$2 approaches $k$3, substantially reducing CPU time with negligible loss of accuracy in the predictable window.
• CNS has been realized for spatiotemporal chaos in systems including the Kuramoto–Sivashinsky, complex Ginzburg–Landau, Rayleigh–Bénard convection, and three-dimensional Kolmogorov turbulence (Qin et al., 15 Jul 2025, Hu et al., 2019, Lin et al., 2016).

6. Impact on Turbulence Simulation and Broader Scientific Context

CNS has provided definitive evidence that traditional DNS is not, by itself, sufficient to guarantee reliable simulation fidelity in chaotic and turbulent flows, even when grid spacing and time-steps appear "fine enough" by conventional metrics. Artificial numerical noise can induce large-scale qualitative changes (e.g., symmetry-breaking, transition to shear/zonal flows, under-prediction of dissipation) and severe statistical errors. CNS has overturned the assumption that finer meshes always yield more accurate DNS for sustained turbulence and has established the necessity of rigorous error control for high-fidelity benchmarking (Qin et al., 2024, Qin et al., 2022, Qin et al., 15 Jul 2025).

CNS has also been instrumental in mathematical analysis, such as the proof of spatial symmetry preservation in the two-dimensional Kolmogorov flow (Liao, 25 Feb 2026). Furthermore, it provides a principled computational platform for investigating mathematical properties of the Navier–Stokes equations, including non-uniqueness and regularity on finite intervals, with direct implications for the Clay Millennium Prize Problem (Liao et al., 13 Feb 2026).

7. Limitations, Scope, and Future Developments

While CNS is the only numerical approach currently capable of resolving chaotic trajectories within any prescribed micro-level uncertainty for large but finite times, its practical limitations are nontrivial:

• Computational costs grow rapidly with the desired horizon of predictability and system size (both $k$4 and $k$5 must be large).
• Multiple-precision arithmetic and high-order recursion scale poorly in high-dimensional problems, restricting CNS to moderate system sizes or requiring extensive parallelization and algorithmic adaptation.
• Shadowing beyond $k$6 is not addressed; CNS certifies trajectories up to, but not beyond, the critical time window dictated by the exponential amplification of noise.
• Extension to non-smooth systems (e.g., discontinuous solutions, shocks) and efficient application to full 3D turbulence at extreme Reynolds numbers requires further methodological advances (Hu et al., 2019, Qin et al., 2023).

Nevertheless, CNS provides an essential toolkit for the scientific and mathematical study of chaos, turbulence, and uncertainty propagation, enabling reliable theoretical and computational investigations where no alternative scheme can currently suffice. CNS also offers a rigorous foundation for validating reduced models, stochastic closures, and data-driven approaches in the presence of intrinsic randomness and extreme sensitivity (Liao et al., 5 Apr 2025, Yang et al., 2021).