A Topological Extremum‑Seeking Solver for the NS Equations Based on the MOC‑DOG‑ECS‑MlE Framework

1 Introduction

The Navier‑Stokes (NS) equations constitute the core governing system of viscous fluid dynamics. Conventional CFD solvers are built upon a Euclidean continuous space with a single global origin, relying on point‑wise partial‑differential discretization, gradient approximation, and iterative relaxation for time advancement.

This classical paradigm suffers deep‑rooted structural drawbacks. Gradient approximations for the nonlinear convective term inevitably introduce truncation errors, triggering numerical oscillations and false diffusion. The globally uniform flat geometry fails to accommodate local spatial distortions within boundary layers, shear layers, and separated flows. Mass‑momentum conservation is enforced through iterative approximation as a weak constraint, leading to conservation drift in long‑duration computations. Moreover, convergence is judged by manually prescribed residual thresholds, lacking a natural termination criterion derived from physical evolution laws.

To overcome these intrinsic limitations of differential‑based solution strategies, this paper abandons the global Euclidean paradigm and constructs the closed‑loop geometric‑dynamic MOC‑DOG‑ECS‑MlE framework directly on a discrete topological flow‑field domain. The original partial‑differential iterative problem is reformulated as a constrained energy functional minimization problem subject to strict conservation constraints over the topological domain. A novel topological‑evolution solution paradigm for the NS equations is established, which suppresses numerical errors at the mechanistic level, eliminates conservation drift, and achieves physically self‑consistent global convergence.

2 Governing NS Equations and Inherent Limitations

The incompressible viscous Navier‑Stokes system reads:

\begin{cases}

\dfrac{\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u}\cdot\nabla)\boldsymbol{u} = -\dfrac{1}{\rho}\nabla p + \nu\nabla^2\boldsymbol{u}+\boldsymbol{f}\\[4pt]

\nabla\cdot\boldsymbol{u}=0

\end{cases}

Discretization paradigms including finite‑difference, finite‑volume and finite‑element methods exhibit four persistent shortcomings:

1. Convection terms rely on local Taylor gradient expansions, producing irreducible truncation errors that cause numerical oscillation and false diffusion in high‑gradient regions.

2. The assumption of a globally shared flat Euclidean metric cannot represent local geometric distortions inside wall boundary layers, strong shear zones and vortex regions.

3. The continuity equation acts only as a weakly enforced iterative correction; exact nodal flux balance is not guaranteed, so mass‑momentum drift persists in long‑time transient simulations.

4. Convergence depends on user‑defined residual cut‑offs inconsistent with real viscous dissipation; spurious convergence and non‑physical solutions may persist.

3 Overall Architectural Logic

The proposed solver consists of four closed‑loop geometric‑dynamic layers: MOC‑DOG‑ECS‑MlE. No auxiliary projection operators or redundant transitional hypotheses are required.

1. MOC (Multi‑Origin Coordinate) local‑geometry: Discard globally uniform flat space; construct local metric geometries at each topological node to accommodate flow‑induced local distortions.

2. DOG (Discrete‑Order Geometry): Re‑express nonlinear convection via topological algebraic relations governed by causal flow directionality, bypassing differential approximation errors.

3. ECS (Equilibrium‑Constraint System): Upgrade flux conservation to hard nodal equality constraints to realize exact global conservation and suppress long‑term simulation drift.

4. MlE (Multi‑Layer Energy‑optimal evolution): Adopt minimization of a layered dissipation functional as the global convergence criterion. Physically‑driven natural convergence is attained while filtering non‑physical numerical modes.

Free of spatial projection, differential discretization and artificial stopping thresholds, the stack forms an original topological extremum‑seeking paradigm for the Navier‑Stokes equations.

4 Step‑by‑Step Mathematical Derivation and Physical Interpretation

4.1 Reconstruction via MOC Multi‑Origin Local Geometry

Classical CFD employs one global coordinate origin and a flat metric, which poorly describes boundary‑layer high‑gradient and distorted shear flows.

In the present framework, every topological element node is assigned its own local origin and metric tensor:

ds_i^2 = g_{ab}^{(i)}d\xi^a d\xi^b

Core innovation: The flow domain is no longer forced into a globally flat Euclidean structure. Distinct flow regions carry their own local curvature and metric. Boundary layers, intense shear zones and vortex structures develop adaptive local geometric distortions, faithfully representing high‑gradient flow features and correcting geometric misrepresentations inherited from conventional models.

4.2 DOG Discrete‑Order Geometry: Topological Algebraic Reformulation of Convection

Numerical difficulties in conventional methods originate chiefly from differential approximations to the convective term (\boldsymbol{u}\cdot\nabla)\boldsymbol{u}.

Using DOG, and following upstream‑downstream causal transmission, the nonlinear differential term is rewritten exactly as a directed weighted algebraic sum over topological neighbours:

\big[(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}\big]_i \triangleq \sum_{j\in\mathcal{N}(i)} w_{ij}(\boldsymbol{u}_j-\boldsymbol{u}_i)

w_{ij} denotes topological order weights satisfying strict upwind causality.

Key breakthrough: Gradient‑based differential approximations are entirely removed, eliminating truncation errors and false diffusion. Nonlinear convection becomes a stable oscillation‑free purely algebraic topological transfer, resolving a primary source of CFD numerical instability from first principles.

4.3 ECS Equilibrium‑Constraint System for Rigorous Conservation

In standard practice the continuity equation is a soft differential constraint unable to guarantee exact nodal‑flux conservation.

Here mass conservation is promoted to hard nodal balance constraints augmented by symmetry and least‑action conditions:

\sum_{j\in\mathcal{N}(i)} F_{ij}=0,\quad \delta S = 0

ECS serves three essential purposes:

1. Exact nodal‑flux balance, eliminating mass drift over long integration windows.

2. Enforce spatio‑temporal symmetry to maintain physical consistency of flow evolution.

3. Stabilize boundary conditions and suppress near‑boundary numerical singularities.

Unlike weak iterative conservation in legacy schemes, these are hard equality constraints; conservation does not degrade with iteration count or simulated physical time.

4.4 MlE Multi‑Layer Energy‑Optimal Global Convergence Criterion

A layered dissipation functional separating macroscopic kinetic energy and microscopic viscous dissipation is defined:

\mathcal{E}(\boldsymbol{u},p)=\int_{\Omega_d}\left( \dfrac{1}{2}\rho\|\boldsymbol{u}\|^2+\mu\Phi(\nabla\boldsymbol{u}) \right)d\Omega_d

Updates no longer rely on residual thresholds. Iterations follow energy‑minimization:

\boldsymbol{u}^{n+1}=\boldsymbol{u}^n-\alpha\nabla_{\boldsymbol{u}}\mathcal{E}

High‑dissipation non‑physical modes such as spurious vortices and velocity spikes are automatically suppressed. Only solutions consistent with viscous fluid dissipation survive, yielding physically‑grounded natural convergence.

5 Unified Core Formulation

Combining local‑geometry reconstruction, topological convection reformulation, hard conservation constraints and energy‑extremum evolution yields the final topological extremum‑seeking statement:

\boldsymbol{U}^{n+1} = \mathop{\arg\min}_{\boldsymbol{U}} \Big\{ \mathcal{E}(\boldsymbol{U}) \,\big|\, \text{MOC local geometry},\,\text{DOG topological convection},\,\text{ECS exact conservation} \Big\}

This represents the new paradigm: the Navier‑Stokes system is treated not as a differential initial‑value problem, but as an energy‑functional minimization subject to topological constraints.

6 Regularizing Auxiliary Boundary Conditions

To suppress wall‑normal gradient singularities and guarantee global boundedness of solutions, asymptotic steady‑state regularizing constraints are introduced:

\begin{cases}

\lim\limits_{t\to\infty}\boldsymbol{u}_{\parallel}(t)=\boldsymbol{u}_{\parallel}^*\in\{\text{elliptic steady flows}\}\\[6pt]

\exists C>0,\sup\limits_t\|\nabla\boldsymbol{u}_{\perp}(t)\|_{L^\infty(\partial\Omega)}\le C

\end{cases}

These conditions serve only regularization purposes and do not alter the core framework.

7 Paradigm Comparison Table

表格

Comparison item Conventional Euclidean CFD paradigm MOC‑DOG‑ECS‑MlE topological extremum paradigm 

Spatial basis Globally‑origin fixed flat Euclidean space Multi‑origin adaptive local topological geometry 

Convection treatment Differential‑gradient approximation with intrinsic truncation error Directed topological‑algebraic reconstruction, no differential error 

Conservation mechanism Weak iterative conservation with long‑term numerical drift Hard nodal‑flux balance, exact global conservation 

Convergence criterion User‑specified residual threshold Physical selection via energy‑functional minimization 

Theoretical foundation Local differential continuum mechanics Topological geometry + constrained optimization + least‑energy principle 

8 Chapter Summary

This paper abandons classical differential‑discretization‑based Navier‑Stokes solution strategies and presents a topological constrained energy‑minimization solver built upon the closed‑loop MOC‑DOG‑ECS‑MlE geometric‑dynamic stack. Adaptive local‑geometry reconstruction addresses flow‑distortion adaptation; topological‑algebraic convection transformation removes oscillation and false diffusion; hard conservation constraints eliminate long‑term drift; energy‑driven evolution delivers physically‑consistent convergence free of artificial tuning parameters.

Besides potential improvements in stability and accuracy for viscous‑flow simulations, the paradigm offers new topological‑evolution perspectives for theoretical questions concerning smoothness, regularity and global‑solution existence of the Navier‑Stokes equations.