arXiv subjects

Klaas Willems

Publications and source records attributed to Klaas Willems.

5 recordsLinked to original sources

arXiv2026

A semi-Lagrangian method for the polyatomic ESBGK model

Polyatomic kinetic models are essential for accurately capturing the thermodynamic behavior of real gases, as internal energy modes significantly influence transport coefficients, relaxation processes, and non-equilibrium effects that cannot be represented by monoatomic models. The polyatomic ESBGK model describes molecular collisions as a relaxation towards a generalized Gaussian distribution with an anisotropic covariance matrix and an exponentially decaying internal energy distribution. We present a new semi-Lagrangian scheme for the polyatomic Ellipsoidal Statistical BGK (ESBGK) model of the Boltzmann equation. The semi-Lagrangian framework, being deterministic and grid-based, removes the time-step restriction associated with the linear transport term by following the method of characteristics. The potentially stiff relaxation term is treated using an implicit A-stable linear multistep method which, owing to the structure of the BGK operator, can be reformulated into a cheap time-stepping scheme. This yields a highly efficient and numerically stable method. The numerical method is asymptotic preserving and stiffly accurate, meaning the scheme asymptotically converges to a scheme for the Euler equations in the vanishing Knudsen limit. In addition, we prove that the first-order scheme, asymptotically converges to the compressible Navier-Stokes equation with correct transport coefficients. Finally, we propose inflow and outflow boundary conditions suitable for BGK-type kinetic equations. We perform simulations of the Fourier and Couette test case to compare the BGK model with Direct Simulation Monte Carlo (DSMC). To conclude, we demonstrate the method on a challenging orifice flow test case with moving boundaries.

math.NA↗

arXiv2026

A meshless MUSCL method for the BGK-Boltzmann equation

We present a numerical method for simulating rarefied gases that interact with moving boundaries and rigid bodies. The gas is described by the BGK equation in Lagrangian form and solved using an Arbitrary Lagrangian-Eulerian method, in which grid points move with the local mean velocity of the gas. The main advantage of the moving grid is that the algorithm can deal well with cases where the domain boundaries are time-dependent and the simulation domain contains rigid objects. Due to the irregular nature of the grid, we use a novel meshless MUSCL-like Moving Least Squares Method (MLS) for spatial discretisation coupled with a higher-order Implicit-Explicit Runge-Kutta method. To avoid spurious oscillations at discontinuities, we use the so-called Multi-dimensional Optimal Order Detection (MOOD) method with an adapted criterion to relax the discrete maximum property. Finally, we employ a new implementation of the boundary conditions that requires no iterative or extrapolation procedure. The method achieves fourth-order in 1D and second-order in 2D for simulations with moving boundaries. We demonstrate the method's effectiveness on classical test cases such as the driven square cavity, shear layer, and shock tube.

math.NA↗

arXiv2025

Higher-order meshless schemes for hyperbolic equations

We discuss the order, efficiency, stability and positivity of several meshless schemes for linear scalar hyperbolic equations. Meshless schemes are Generalised Finite Difference Methods (GFDMs) for arbitrary irregular grids in which there is no connectivity between the grid points. We propose a new MUSCL-like meshless scheme that uses a central stencil, with which we can achieve arbitrarily high orders, and compare it to existing meshless upwind schemes and meshless WENO schemes. The stability of the newly proposed scheme is guaranteed by an upwind reconstruction to the midpoints of the stencil. The new meshless MUSCL scheme is also efficient due to the reuse of the GFDM solution in the reconstruction. We combine the new MUSCL scheme with a Multi-dimensional Optimal Order Detection (MOOD) procedure to avoid spurious oscillations at discontinuities. In one spatial dimension, our fourth order MUSCL scheme outperforms existing WENO and upwind schemes in terms of stability and accuracy. In two spatial dimensions, our MUSCL scheme achieves similar accuracy to an existing WENO scheme but is significantly more stable.

math.NA↗

arXiv2025

Kinetic-Diffusion-Rotation Algorithm for Dose Estimation in Electron Beam Therapy

Monte Carlo methods are state-of-the-art when it comes to dosimetric computations in radiotherapy. However, the execution time of these methods suffers in high-collisional regimes. We address this problem by introducing a kinetic-diffusion particle tracing scheme. This algorithm, first proposed in the context of neutral transport in fusion energy, relies on explicit simulation of the kinetic motion in low-collisional regimes and dynamically switches to motion based on a random walk in high-collisional regimes. The random walk motion maintains the first two moments (mean and variance) of the kinetic motion. We derive an analytic formula for the mean kinetic motion and discuss the addition of a multiple scattering distribution to the algorithm. In contrast to neutral transport, the radiation transfer setting does not readily admit to an analytical expression for the variance of the kinetic motion, and we therefore resort to the use of a lookup table. We test the algorithm for dosimetric computations in radiation therapy on a 2D CT scan of a lung patient. Using a simple particle model, our Python implementation of the algorithm is nearly 33 times faster than an equivalent kinetic simulation at the cost of a small modeling error.

q-bio.QM↗

arXiv2024

High-performance computing for the BGK model of the Boltzmann equation with a meshfree Arbitrary Lagrangian-Eulerian (ALE) method

In this paper, we present high-performance computing for the BGK model of the Boltzmann equation with a mesh-free method. For the numerical simulation of the BGK equation we use an Arbitrary-Lagrangian-Eulerian (ALE) method developed in previous work, where the approximation of spatial derivatives and the reconstruction of a function is based on the weighted least squares method. A Graphics Processing Unit (GPU) is used to accelerate the code. The result is compared with sequential and parallel versions of the CPU code. Two and three-dimensional driven cavity problems are solved, where a speed-up of several orders of magnitude is obtained compared to a sequential CPU simulation.

math.NA↗