arXiv subjects

Eugene de Villiers

Publications and source records attributed to Eugene de Villiers.

6 recordsLinked to original sources

arXiv2026

Quantum Algorithms for Computational Fluid Dynamics

We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow--Hassidim--Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum--classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.

quant-ph↗

arXiv2026

Quantum-Inspired Simulation of 2D Turbulent Rayleigh-Bénard Convection

Turbulent thermal convection governs heat transport in systems ranging from stellar interiors to industrial heat exchangers. Two-dimensional Rayleigh-Bénard convection serves as a paradigm for these flows, reproducing key features such as thin boundary layers, large-scale circulation, and sustained plume dynamics. While Matrix Product State (MPS) methods have demonstrated significant compression of isothermal turbulent fields, their application to buoyancy-driven flows with active thermal coupling has remained unexplored. We apply MPS to two-dimensional Rayleigh-Bénard convection with dynamical simulations up to $\mathrm{Ra} = 10^{10}$. An a priori decomposition of DNS snapshots up to $\mathrm{Ra} = 10^{11}$ shows that the bond dimension $χ$ required to represent the flow fields grows without saturation, in contrast to the plateauing of $χ$ reported for velocity fields in isothermal 2D turbulence. Crucially, however, dynamical simulations solving the governing equations directly in the compressed MPS format at fixed $χ$ show that the $χ$ required to recover statistical observables, such as the Nusselt number, scales significantly more favorably with $\mathrm{Ra}$ than the a priori complexity suggests. At $\mathrm{Ra} = 10^{10}$, a relative error of $1.8\%$ in the mean Nusselt number is achieved with a nearly 9-fold reduction in degrees of freedom, using a $χ$ comparable to that required at $\mathrm{Ra} = 10^{9}$. Spectral analysis confirms the progressive recovery of spatial and temporal scales with increasing $χ$. These findings establish MPS as a scalable tool for simulating thermally driven turbulence, suggesting the method may remain viable for investigations of the ultimate regime at substantially higher $\mathrm{Ra}$.

physics.flu-dyn↗

arXiv2025

A Framework for Initial Transient Detection and Statistical Assessment of Convergence in CFD Simulations

Time series data often contain initial transient periods before reaching a stable state, posing challenges in analysis and interpretation. In this paper, we propose a novel approach to detect and estimate the end of the initial transient in time series data. Our method leverages the reversal mean standard error (RMSE) as a metric for assessing the stability of the data. Additionally, we employ fractional filtering techniques to enhance the detection accuracy by filtering out noise and capturing essential features of the underlying dynamics. Combining with autocorrelation-corrected confidence intervals we provide a robust framework to automate transient detection and convergence assessment. The method ensures statistical rigor by accounting for autocorrelation effects, validated through simulations with varying time steps. Results demonstrate independence from numerical parameters (e.g., time step size, under-relaxation factors), offering a reliable tool for steady-state analysis. The framework is lightweight, generalizable, and mitigates inflated false positives in autocorrelated datasets.

stat.ME↗

arXiv2024

Boundary Treatment for Variational Quantum Simulations of Partial Differential Equations on Quantum Computers

The paper presents a variational quantum algorithm to solve initial-boundary value problems described by second-order partial differential equations. The approach uses hybrid classical/quantum hardware that is well suited for quantum computers of the current noisy intermediate-scale quantum era. The partial differential equation is initially translated into an optimal control problem with a modular control-to-state operator (ansatz). The objective function and its derivatives required by the optimizer can efficiently be evaluated on a quantum computer by measuring an ancilla qubit, while the optimization procedure employs classical hardware. The focal aspect of the study is the treatment of boundary conditions, which is tailored to the properties of the quantum hardware using a correction technique. For this purpose, the boundary conditions and the discretized terms of the partial differential equation are decomposed into a sequence of unitary operations and subsequently compiled into quantum gates. The accuracy and gate complexity of the approach are assessed for second-order partial differential equations by classically emulating the quantum hardware. The examples include steady and unsteady diffusive transport equations for a scalar property in combination with various Dirichlet, Neumann, or Robin conditions. The results of this flexible approach display a robust behavior and a strong predictive accuracy in combination with a remarkable polylog complexity scaling in the number of qubits of the involved quantum circuits. Remaining challenges refer to adaptive ansatz strategies that speed up the optimization procedure.

quant-ph↗

arXiv2021

A unification of least-squares and Green-Gauss gradients under a common projection-based gradient reconstruction framework

We propose a family of gradient reconstruction schemes based on the solution of over-determined systems by orthogonal or oblique projections. In the case of orthogonal projections, we retrieve familiar weighted least-squares gradients, but we also propose new direction-weighted variants. On the other hand, using oblique projections that employ cell face normal vectors we derive variations of consistent Green-Gauss gradients, which we call Taylor-Gauss gradients. The gradients are tested and compared on a variety of grids such as structured, locally refined, randomly perturbed, unstructured, and with high aspect ratio. The tests include quadrilateral and triangular grids, and employ both compact and extended stencils, and observations are made about the best choice of gradient and weighting scheme for each case. On high aspect ratio grids, it is found that most gradients can exhibit a kind of numerical instability that may be so severe as to make the gradient unusable. A theoretical analysis of the instability reveals that it is triggered by roundoff errors in the calculation of the cell centroids, but ultimately is due to truncation errors of the gradient reconstruction scheme, rather than roundoff errors. Based on this analysis, we provide guidelines on the range of weights that can be used safely with least squares methods to avoid this instability.

physics.comp-ph↗

arXiv2019

A family of first-order accurate gradient schemes for finite volume methods

A new discretisation scheme for the gradient operator, suitable for use in second-order accurate Finite Volume Methods (FVMs), is proposed. The derivation of this scheme, which we call the Taylor-Gauss (TG) gradient, is similar to that of the least-squares (LS) gradients, whereby the values of the differentiated variable at neighbouring cell centres are expanded in truncated Taylor series about the centre of the current cell, and the resulting equations are summed after being weighted by chosen vectors. Unlike in the LS gradients, the TG gradients use vectors aligned with the face normals, resembling the Green-Gauss (GG) gradients in this respect. Thus, the TG and LS gradients belong in a general unified framework, within which other gradients can also be derived. The similarity with the LS gradients allows us to try different weighting schemes (magnitudes of the weighting vectors) such as weighting by inverse distance or face area. The TG gradients are tested on a variety of grids such as structured, locally refined, randomly perturbed, and with high aspect ratio. They are shown to be at least first-order accurate in all cases, and are thus suitable for use in second-order accurate FVMs. In many cases they compare favourably over existing schemes.

math.NA↗