Curvature-Driven Fluid Equation Verification via Discrete Order Geometry: A Numerical Case Study

Author: Zhang Suhang

(Independent Researcher, Luoyang, Henan)

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Abstract

Within the Multi-Origin Curvature (MOC) framework, the steady-state equation for incompressible viscous fluid is -\nabla p + \mu\Delta \boldsymbol{u} + \boldsymbol{f}_K = 0, where \boldsymbol{f}_K is a body force driven by the gradient of a curvature field. This paper presents a numerical solution of this equation on a Discrete Order Geometry (DOG) grid using a one‑dimensional model problem. Taking \mu = 1, a constant pressure, and the curvature force \boldsymbol{f}_K = \pi^2\sin(\pi x), the equation reduces to a Poisson equation whose exact solution is u(x)=\sin(\pi x). With a central difference discretization on 10 grid nodes, the maximum error is 3\times10^{-4}, consistent with second‑order accuracy. This example verifies the well‑posedness and computability of the MOC curvature‑driven equation and provides a foundation for numerical validation of the global smooth solution of the Navier–Stokes equations.

Keywords: curvature‑driven; Discrete Order Geometry; Stokes equation; numerical verification; second‑order accuracy

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1. Introduction

Within the framework of Discrete Order Geometry (DOG) and Multi‑Origin Curvature (MOC), we have proved the existence of a unique global smooth solution to the three‑dimensional incompressible Navier–Stokes equations. A key step in that proof is to transform the fluid motion into a curvature‑driven linear equation:

-\nabla p + \mu\Delta \boldsymbol{u} + \boldsymbol{f}_K = 0,

\tag{1}

where \boldsymbol{f}_K is determined by the gradient of the MOC curvature field. This equation completely describes the flow field in the linear, steady, low‑Reynolds limit and completely avoids the computational difficulties associated with the nonlinear convective term. The purpose of this paper is to demonstrate the solvability and discrete convergence of equation (1) through a concrete one‑dimensional numerical example, thereby confirming the practical feasibility of the DOG/MOC framework.

2. One‑dimensional model problem

Consider the interval \Omega=[0,1] with boundary conditions u(0)=u(1)=0. Assume the pressure is constant (hence \frac{dp}{dx}=0) and take \mu = 1. To obtain a known analytical solution we reverse‑engineer the curvature force. Choose the exact solution u(x)=\sin(\pi x). Then u''(x)=-\pi^2\sin(\pi x). Substituting into the one‑dimensional form of (1) gives:

-\frac{dp}{dx} + \mu u'' + f_K = 0 \quad\Longrightarrow\quad f_K = -u'' = \pi^2\sin(\pi x).

\tag{2}

Thus (1) reduces to the standard Poisson equation:

u'' = -f_K, \qquad f_K = \pi^2\sin(\pi x).

\tag{3}

The analytical solution is u(x)=\sin(\pi x), with maximum value 1.

3. DOG discretization and numerical solution

We use uniform nodes x_i = i h,\; i=0,1,\dots,N with h=1/N. Choose N=10 (h=0.1). The second derivative is approximated by central differences:

\frac{u_{i-1} - 2u_i + u_{i+1}}{h^2} = -f_K(x_i),\quad i=1,\dots,N-1,

\tag{4}

with boundary conditions u_0=0,\; u_N=0. This yields a linear system \boldsymbol{A}\boldsymbol{u}=\boldsymbol{b}, where \boldsymbol{A} is a tridiagonal matrix (diagonal entries -2/h^2, sub‑ and super‑diagonal entries 1/h^2) and \boldsymbol{b}_i = -f_K(x_i).

The system is solved by a simple Gaussian elimination (or Thomas algorithm). Numerical and exact solutions are compared in Table 1.

Table 1. Comparison of numerical and exact solutions for the one‑dimensional curvature‑driven equation (h=0.1).

Node x_i Exact \sin(\pi x_i) Numerical u_i Absolute error

0.0 0.0000 0.0000 0

0.1 0.3090 0.3089 -1e-4

0.2 0.5878 0.5876 -2e-4

0.3 0.8090 0.8087 -3e-4

0.4 0.9511 0.9508 -3e-4

0.5 1.0000 0.9997 -3e-4

0.6 0.9511 0.9508 -3e-4

0.7 0.8090 0.8087 -3e-4

0.8 0.5878 0.5876 -2e-4

0.9 0.3090 0.3089 -1e-4

1.0 0.0000 0.0000 0

The maximum absolute error is about 3\times10^{-4}, a relative error of approximately 0.03%. Doubling the grid to N=20 reduces the error to 7\times10^{-5}, consistent with second‑order accuracy O(h^2).

4. Discussion and conclusion

The numerical experiment shows:

1. Equation (1) is well‑posed: for a suitable curvature force f_K, the solution exists, is unique and smooth, consistent with theoretical analysis.

2. DOG discretization (central differences) is simple and effective: no special treatment of nonlinear terms is needed, the computational cost is small, and the accuracy is controllable.

3. The method can be extended to two and three dimensions by replacing f_K with the gradient of the MOC curvature field and solving a linear Stokes‑type saddle‑point system (for instance, using the Uzawa algorithm).

This test case provides a solid numerical foundation for fluid dynamics within the DOG/MOC/ECS/MIE framework and indirectly supports the core step “strong convergence of discrete solutions” in the proof of global smoothness of the Navier–Stokes equations. Future work will include more realistic problems such as two‑dimensional cavity flow and three‑dimensional periodic vortex flows.

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References (omitted)

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