Methods for Computational Fluid Dynamics Aiming Quantum Computing

Resumo

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⟩ + b|ψ₂⟩, (1) where U is a unitary operator (U^†U = I), |ψ_i⟩ 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]: u_analytical = [formula omitted in abstract summary]. 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: u_approx = U_:,1:r · S_1:r,1:r · V^T_1:r,:, (5) where U, S, and V are singular vectors and values, and r is the rank. The linearized fields are updated as: u_linearized = u_approx − αu_approxΔt, (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]: u_log = log(|u| + ϵ), u_linearized = exp (u_log − αu_logΔt) · 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. [...]