High Performance Computational Methods for the Boltzmann Equation

Project Information

HYPERBOLE

Grant agreement ID: 101105786

DOI

10.3030/101105786

Project closed

EC signature date 26 April 2023

Start date 1 January 2024

End date 31 May 2026

Funded under

• Marie Skłodowska-Curie Actions (MSCA)

Total cost

No data

EU contribution

€ 203 464,32

Coordinated by

TECHNISCHE UNIVERSITEIT EINDHOVEN
Netherlands

Periodic Reporting for period 1 - HYPERBOLE (High Performance Computational Methods for the Boltzmann Equation)

Reporting period: 2024-01-01 to 2026-05-31

Summary of the context and overall objectives of the project

Making computer chips is extraordinarily difficult. The machines that print the finest circuitry use extreme ultraviolet light, which is absorbed by almost anything it touches, so the entire optical path must be held at near-vacuum. A directed flow of inert gas carries away heat and keeps debris from the surfaces that matter. At those pressures the gas no longer behaves like a continuous fluid: molecules travel comparatively far between collisions, and the equations that describe air flowing over a wing simply cease to apply.

The same regime appears elsewhere. Satellites in very low orbits skim through an atmosphere too thin to be a fluid yet dense enough to slow them down. Fusion devices, vacuum coating systems and spacecraft re-entry pose the same question: how does a gas behave when its molecules meet only rarely?

The equation that answers it is the Boltzmann equation, valid across the whole range from dense to nearly collisionless. Its power comes at a price. Rather than tracking a few averaged quantities at each point in space, it tracks how molecular velocities are distributed, placing the unknown in a seven-dimensional space of position, velocity and time. Discretising even a modest problem produces billions of unknowns before any physics is computed.

Industry's usual answer is a stochastic method that represents the gas by simulated particles and lets chance decide their collisions. It is robust and mature, but built on random sampling, so its output carries statistical noise that falls only as the square root of the number of samples. Suppressing that noise to a level useful for design can be prohibitively expensive, and the difficulty is worst exactly where demand is growing: slow, subtle, unsteady flows in which the signal of interest is small, and coupled problems where a noisy answer cannot be fed cleanly into a deterministic structural or optical solver.

Deterministic methods produce no noise at all, but historically they have been too costly for three-dimensional use. HYPERBOLE set out to change that balance — not by demanding more computing power, but by finding and exploiting mathematical structure hidden inside the equations, and by designing software able to use modern hardware efficiently. The aim is to move noise-free simulation of rarefied gas from a research curiosity towards a practical engineering tool for the high-tech sectors, from semiconductor manufacturing to space, on which European technological autonomy increasingly depends.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The project concentrated on the most expensive component of the Boltzmann equation: the collision term, which describes what happens when molecules meet. In the spectral approach used here it is represented by a large three-index tensor whose size grows as the sixth power of the angular resolution — the reason such solvers have been confined to coarse resolutions and modest problems.

The central achievement is the demonstration that most of that cost is not physics but bookkeeping. A collision does not care how the observer is oriented: rotate the whole configuration and the outcome is identical. Conventional formulations nonetheless assemble the collision geometry in a fixed laboratory frame, recomputing the same physics in every orientation.

The project made this insight exact and computable. By rigidly rotating the reference frame to align with the colliding pair and integrating analytically over the full group of three-dimensional rotations, the eight-dimensional integral defining the collision tensor reduces exactly — with no approximation — to a five-dimensional kinematic core. What remains separates cleanly: the angular geometry, which has a closed-form algebraic expression and is evaluated exactly, and the scattering physics, computed numerically. This separation is a Wigner–Eckart factorisation, bringing machinery long used in quantum mechanics to bear on rarefied gas dynamics.

Three further developments turned the factorisation into working software. A tailored integration strategy resolves a mathematical singularity in the collision kernel by splitting and remapping the integration domain, so the scattering physics converges exponentially and is evaluated to machine precision. A compressed storage format exploits the fact that the overwhelming majority of angular interactions are forbidden by symmetry, storing only the survivors as a compact routing table. And the physical conservation laws — mass, momentum and energy — are guaranteed by construction, by zeroing precisely identified entries of the precomputed tensor, so conservation holds to machine precision at no runtime cost.

The measured effect is substantial. Arithmetic cost falls from the sixth to the fifth power of angular resolution, and a cache-optimised contraction strategy delivers a 37-fold single-core speedup over the standard dense formulation. Memory requirements fall by three orders of magnitude, from tens of gigabytes to tens of megabytes at practical resolutions.

The implementation was validated against every analytical benchmark available in the field: an exact transient solution for Maxwell molecules, the classical eigenvalue spectrum of the linearised operator, and the infinite-order transport coefficients for hard spheres, each reproduced to the precision of the arithmetic, with the conservation laws and equilibrium state preserved exactly and the second law respected. The outcome of the action is a validated, openly available computational method that removes the memory and cost barriers which had kept high-accuracy deterministic Boltzmann solvers from three-dimensional use.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Spectral methods for the Boltzmann collision operator were already known to offer high accuracy and exact conservation, but their memory and arithmetic demands had kept them from practical three-dimensional use. Removing three orders of magnitude of storage and a full power of the resolution scaling changes what is reachable on ordinary hardware, and does so without approximation: the factorisation is exact, and the conservation properties are stronger than before rather than traded away.

The potential impacts follow from the physics. In chip manufacturing, noise-free simulation of low-pressure gas flow and debris transport supports the design of cleaner, more reliable lithography systems, in a supply chain Europe has identified as strategically critical. In space, accurate modelling of extremely thin atmospheres bears on satellites in very low orbits, where atmospheric drag is both the principal obstacle and, through air-breathing propulsion concepts, a potential resource; such orbits give sharper observation and lower latency, and residual drag removes defunct hardware naturally rather than adding to long-lived debris. In fusion and vacuum technology, the same physics governs how gases and impurities move through low-pressure systems.

Realising these impacts requires carrying the method from validated research code towards production engineering tools, and connecting it to the proprietary problem geometries of individual industries. That route is already open: the methods form the technical foundation of Simkinetic, a Dutch deep-tech company building fast, accurate simulation tools for extreme gas environments, with ongoing collaborations with industrial and research partners in semiconductor manufacturing and aerospace. Key needs for further uptake are continued engineering development towards three-dimensional industrial cases, access to high-performance computing at scale, and sustained access to markets and finance to mature the technology from its current readiness level towards deployment. The underlying theory and software have been placed in the public domain through open-access publication and open-source release, so the scientific advance remains available to the whole research community independently of any commercial development built upon it.