Methods for Computer Fluid Dynamics aiming Quantum Computing

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

Methods for Computational Fluid Dynamics Aiming Quantum Computing
Thiago F. P. O. Chahin¹, Leonardo R. Monteiro²
LACIA/UDESC, Joinville, SC

This work investigates linearization techniques for the Navier-Stokes equations (NSE) using the Taylor-Green Vortex (TGV) benchmark, with applications to Quantum Computing (QC). Pre-processing NSE for QC will open avenues for faster fluid simulations in the future [4].

Quantum algorithms require linear formulations, as quantum operations follow the superposition principle through unitary transformations [2]. This fundament is shown in Dirac notation:

  U (a|ψ₁⟩ + b|ψ₂⟩) = aU|ψ₁⟩ + bU|ψ₂⟩, (1)

where U is a unitary operator (U†U = I), |ψᵢ⟩ represent quantum state vectors, and a, b ∈ ℂ are complex probability amplitudes. The nonlinear convective term (u · ∇)u in NSE violates this linearity requirement, necessitating specialized approximation techniques. The TGV problem provides an ideal test case for its exact analytical solution and periodic boundary conditions [1, 7].

The TGV test case is governed by the incompressible NSE [7]:

  ∂u/∂t + (u · ∇)u = −∇p + ν∇²u,  ∇ · u = 0, (2)

where u is the velocity field, p is the pressure, ν is the kinematic viscosity, and t is time.

The analytical solution for TGV is [7]:

  uₐₙₐₗᵧₜᵢcₐₗ =
  (
   A cos(ax) sin(by) sin(cz) e^(−ν(a² + b² + c²)t
   B sin(ax) cos(by) sin(cz) e^(−ν(a² + b² + c²)t
   C sin(ax) sin(by) cos(cz) e^(−ν(a² + b² + c²)t
  ), (3)

where A, B, C, a, b, and c are constants, and x, y, and z are Cartesian coordinates.

The studied methods are:

1. Local Temporal Linearization: Approximates the nonlinear term using a Taylor series expansion around equilibrium u₀ [6]:

  (u · ∇)u ≈ (u₀ · ∇)u₀ + α[(u₀ · ∇)δu + (δu · ∇)u₀], (4)

where δu = u − u₀ is a small perturbation, and α is a coefficient that calibrates the velocity field at each time step.

2. SVD Matricial Tensorial Linearization: Uses Singular Value Decomposition (SVD) to approximate velocity fields in low-rank format, reducing complexity while preserving flow features. The 3D fields are reshaped into 2D matrices, decomposed via SVD, and rebuilt using dominant singular components:

  uₐₚₚᵣₒₓ = U[:,1:r] · S[1:r,1:r] · Vᵀ[1:r,:], (5)

where U, S, and V are singular vectors and values, and r is the rank. The linearized fields are updated as:

  uₗᵢₙₑₐᵣᵢzed = uₐₚₚᵣₒₓ − α uₐₚₚᵣₒₓ Δt, (6)

where α is a linearization coefficient and Δt is the time step. This approach aligns with low-rank solvers for Navier-Stokes equations [3].

3. Logarithmic Linearization: This method linearizes nonlinear velocity terms by applying a logarithmic transformation to |u|, ensuring positivity with a small constant ϵ = 10⁻¹⁰. Linearization is performed in log-space and mapped back via the exponential function, preserving velocity direction [5]:

  uₗₒg = log(|u| + ϵ), uₗᵢₙₑₐᵣᵢzed = exp(uₗₒg − α uₗₒg Δt) · (u / |u|), (7)

where α is a linearization coefficient and Δt the time step.

Numerical experiments used a 64³ grid, 2π domain, ν = 0.01 m²/s, Δt = 0.001 s, third-order Runge-Kutta time integration [1], and finite-differences for spatial derivatives.

The methods yielded comparable accuracy to standard NSE:

Navier-Stokes:     MSE = 0.049062, Absolute Error = 0.161511
Local Temporal:  MSE = 0.049062, Absolute Error = 0.161511
SVD:         MSE = 0.049001, Absolute Error = 0.161220
Logarithmic:      MSE = 0.049049, Absolute Error = 0.161493

The SVD method showed superior performance, demonstrating potential for quantum computing applications where linear formulations are essential.

References
[1] M. E. Brachet, D. I. Meiron, S. A. Orszag, B. G. Nickel, R. H. Morf, and U. Frisch. “Small-scale structure of the Taylor-Green vortex”. Journal of Fluid Mechanics 130 (1983), pp. 411–452. doi: 10.1017/S0022112083001159.
[2] A. W. Harrow. “Quantum Algorithms for Systems of Linear Equations”. arXiv preprint (2015). https://arxiv.org/abs/1501.0008.
[3] K. Lee, H. C. Elman, and B. Sousedík. “A Low-Rank Solver for the Navier-Stokes Equations with Uncertain Viscosity”. SIAM Journal on Scientific Computing 41.5 (2019), A1277–A1304. doi: 10.1137/18M1186801.
[4] X. Li, X. Yin, N. Wiebe, J. Chun, G. K. Schenter, M. S. Cheung, and J. Millmenstadt. “Potential quantum advantage for simulation of fluid dynamics”. Physical Review Research 7.1 (2025), p. 013036. doi: 10.1103/PhysRevResearch.7.013036.
[5] M. Ltifi. “Strong solution of modified 3D-Navier-Stokes equations”. arXiv preprint (2021). https://arxiv.org/abs/2111.00859.
[6] T. W. H. Sheu and R. K. Lin. “Newton linearization of the incompressible Navier-Stokes equations”. International Journal for Numerical Methods in Fluids 44.3 (2004), pp. 297–312. doi: 10.1002/fld.639.
[7] G. I. Taylor and A. E. Green. “Mechanism of the production of small eddies from large ones”. Proceedings of the Royal Society of London. Series A 158.895 (1937), pp. 499–521. doi: 10.1098/rspa.1937.0035.

¹ [email protected]  
² [email protected]