3D Incompressible Navier–Stokes in a Periodic Box Finite Difference, Quantum-Inspired Finite Difference, Crank–Nicolson, and Quantum-Inspired Crank–Nicolson

Abstract

This paper demonstrates four numerical solvers for the 3D incompressible Navier–Stokes equations on a periodic cubic domain: (a) explicit finite difference (FD), (b) quantum-inspired (QI) FD, (c) semi-implicit Crank–Nicolson (CN) (diffusion treated with CN), and (d) QI-CN. All methods use a projection (fractional-step) approach to enforce incompressibility by solving a pressure Poisson equation. We select a standard periodic benchmark (Taylor–Green vortex) and show how the four schemes evolve velocity magnitude, vorticity magnitude, and kinetic energy decay. The “quantum-inspired” variants are implemented as local unitary-like amplitude rotations between velocity components driven by vorticity, which preserves local kinetic energy while adding structured mixing. The complete Python code (NumPy + Matplotlib only) is provided and generates multiple colorful figures for comparison. Projection ideas follow classical formulations introduced by Chorin and Temam, with modern second-order projection refinements widely studied in CFD literature.