arXiv subjects

Manuel Torrilhon

Publications and source records attributed to Manuel Torrilhon.

At least 19 recordsLinked to original sources

arXiv2026

Revisiting the Coupling of Thermodynamics and Electromagnetics

We revisit the coupling of continuum thermodynamics and electromagnetic theory for polarisable and magnetisable matter in motion. Two routes are followed and then compared. The first route is the axiomatic bulk theory of Dreyer, Guhlke and M\"uller, in which universal balance laws are closed by an entropy principle. We show that the source of the internal energy balance must be built with the non-convective electric current, that the polarisation current and the Lorentz magnetisation enter through one single identity, which Dreyer et al.\ do not write down, and that this identity fixes both the admissible entropy variables and the signs of the bound-current ansatz. The second route is the statistical-mechanical one of Mazur, in which the macroscopic Maxwell equations are obtained by ensemble averaging over a system of atoms with internal charge carriers. Mazur stops before the conservation laws, so we derive them, and we estimate the size of the mass-correction terms that appear. The comparison shows that after a redefinition of polarisation and magnetisation the two sets of equations agree structurally. The only irreducible difference is a momentum contribution from microscopic field fluctuations, which can not be reproduced in a purely macroscopic theory. We further show that the electromotive intensity $\mathcal{E}$ and the Lorentz magnetisation $\mathcal{M}$ are not modelling choices but appear by themselves, and that the asymmetric look of the entropy function is a consequence of the chosen energy variable and not a defect of the theory.

cond-mat.stat-mech↗

arXiv2026

A Splitting Scheme for Dispersive Shallow Moment Equations

The well-known Shallow Water Equations (SWE) are used for modeling incompressible free-surface flows whenever the shallowness allows for a vertical-averaging; i.e., vertical effects are negligible in comparison to horizontal ones. But vertical averaging comes with the price of losing information along the vertical axis. Moment models for shallow flow contain information on the vertical velocity and pressure profile despite being dimensionally reduced. A class of these models incorporating a non-hydrostatic pressure have been introduced before as Dispersive Shallow Moment Models (DSM). However, no method for solving the non-stationary equations has been presented yet, mainly because it was unclear how to compute the pressure equation in the form of the divergence-free constraint. We rewrite the pressure equations of the DSM models in the form of a Poisson-like problem to enable their solution with a projection-type splitting scheme. For the linear equations, we present the calculations for the generalized model and discuss the non-linear case. We state the first two linear models and the corresponding nonlinear counterparts. Finally, we introduce a hybrid Finite-Volume Finite-Difference method and discuss the non-stationary numerical results for an experiment with periodic boundary and uneven bottom topography.

physics.flu-dyn↗

arXiv2026

Sparse and low-rank kinetic distribution estimation

In this paper, we consider methods that allow for memory-efficient storage of high-dimensional distributions and retain certain key features thereof, specifically in a kinetic theory context. We propose an extension to the entropic quadrature method that allows for enforcing sparsity, and propose a new low-rank decomposition approach that ensures preservation of moment information. The methods are applied to several model kinetic distributions, as well as to distributions obtained from high-resolution kinetic simulations of the Vlasov--Maxwell system.

physics.comp-ph↗

arXiv2026

Moment-preserving particle merging via non-negative least squares

A novel particle merging algorithm for rarefied gas dynamics simulations is proposed that can conserve arbitrary velocity and spatial moments of the particle distribution via solving a non-negative least squares problem. An extension that preserves both exact and approximate collision rates is also derived. The algorithm is applied to the simulation of several model rarefied gas dynamics problems, where it exhibits noticeably lower merging-induced error in key macroscopic quantities.

physics.comp-ph↗

arXiv2026

Volume Term Adaptivity for Discontinuous Galerkin Schemes

We introduce the concept of volume term adaptivity for high-order discontinuous Galerkin (DG) schemes solving time-dependent partial differential equations. Termed v-adaptivity, we present a novel general approach that exchanges the discretization of the volume contribution of the DG scheme at every Runge-Kutta stage based on suitable indicators. Depending on whether robustness or efficiency is the main concern, different adaptation strategies can be chosen. Precisely, the weak form volume term discretization is used instead of the entropy-conserving flux-differencing volume integral whenever the former produces more entropy than the latter, resulting in an entropy-stable scheme. Conversely, if increasing the efficiency is the main objective, the weak form volume integral may be employed as long as it does not increase entropy beyond a certain threshold or cause instabilities. Thus, depending on the choice of the indicator, the v-adaptive DG scheme improves robustness, efficiency and approximation quality compared to schemes with a uniform volume term discretization. We thoroughly verify the accuracy, linear stability, and entropy-admissibility of the v-adaptive DG scheme before applying it to various compressible flow problems in two and three dimensions.

math.NA↗

arXiv2026

Variance Reduction in the Fokker-Planck Particle Method for Rarefied Gases using Quasi-Random Numbers

The Fokker-Planck (FP) particle method accelerates rarefied-gas simulations by replacing the binary collisions of the commonly used Direct Simulation Monte Carlo (DSMC) method with a drift=diffusion process. Like all particle methods, the FP method is inherently stochastic, which leads to statistical fluctuations in macroscopic quantities and necessitates large particle numbers for accurate results. In this work, we investigate the use of quasi-random numbers, which sample distributions more evenly and thereby reduce the variance. To preserve the low-discrepancy structure across time steps, we employ the Array Randomized Quasi-Monte Carlo (Array-RQMC) technique. We combine the FP method with Array-RQMC and compare it in homogeneous and inhomogeneous problems with other commonly used variance-reduction techniques. The proposed FP-Array-RQMC approach achieves improved convergence rates compared with pseudo-random sampling and yields smaller estimator errors for sufficiently large particle numbers.

math.NA↗

arXiv2025

Moment Approximations to Magnetic Rotating Shallow Flows

Originally introduced to describe a transition region in stars, the magnetic rotating shallow water (MRSW) model is now used in many solar physics and geophysical applications. Derived from the 3-D incompressible magnetohydrodynamic system, the shallow nature of these applications motivates depth-averaging of both the velocities and magnetic fields. This is advantageous in terms of computational efficiency -- but at the loss of vertical information, thus limiting the predictive power of the MRSW model. To overcome this problem, we employ higher-order vertical moments, but now in the context of conductive fluids. In doing so, the new approximation maintains non-constant vertical profiles of both the horizontal magnetic fields and horizontal velocities, while still remaining in the simplified 2-D framework corresponding to depth integration. In this work, we extend the derivation of the shallow water moment equations to derive the MRSW moment system of arbitrary order; i.e., we represent the vertical profiles of the velocities -- and now additionally the magnetic fields -- by arbitrary-order polynomial expansions, and close the new expanded 2-D system with evolution equations for these polynomial coefficients, found via Galerkin projection. Through numerical experiments for MRSW moment systems up to third-order, we demonstrate that these moment approximations reduce model error without significantly sacrificing computational efficiency.

math.NA↗

arXiv2025

Extending the Numerical Flow Iteration to the multi-species Vlasov-Maxwell system through Hamiltonian Splitting

The Numerical Flow Iteration (NuFI) method has recently been proposed as a memory-slim while accurate in phase-space method for the electro-static Vlasov--Poisson system. It stores the temporal evolution of the electric field, instead of the distribution functions, and reconstructs the solution in each time step by following the characteristics backwards in time and reconstructing the solution from the initial distribution. NuFI has been shown to be more accurate than other state-of-the-art electro-static Vlasov solvers given the same amount of degrees of freedom. In this paper, we build on the Hamiltonian structure of the full Vlasov--Maxwell system to extend NuFI to handle electro-magnetic kinetic plasma dynamics. We show that the structure-preserving properties of the NuFI time-stepping are preserved when extending to the electro-magnetic case. Furthermore we discuss how NuFI can be incorporated into existing Semi-Lagrangian codes as an efficient while accurate subcycling technique.

physics.plasm-ph↗

arXiv2025

Simulation of multi-species kinetic instabilities with the Numerical Flow Iteration

Kinetic instabilities are one of the most challenging aspects in computational plasma physics. Accurately capturing their onset and evolution requires fine resolution of the high-dimensional distribution functions of each relevant species, which quickly becomes computationally prohibitively expensive. Additionally, plasma dynamics is an inherently multi-scale phenomenon due to the vast separation of scales between heavy ions and light-weight electrons. In previous work the Numerical Flow Iteration (NuFI) was suggested as a high-fidelity alternative with reduced memory complexity making it an interesting candidate to simulate complicated kinetic instabilities. In this work we extend NuFI to non-periodic boundary conditions and demonstrate how it is possible to reduce the computational complexity to allow for longer simulation periods.

physics.plasm-ph↗

arXiv2025

Extraction of Moment Closures for Strongly Non-Equilibrium Flows via Machine Learning

We introduce a machine learning framework for moment-equation modeling of rarefied gas flows, addressing strongly non-equilibrium conditions inaccessible to conventional computational fluid dynamics. Our approach utilizes high-order moments and collision integrals, highly sensitive to non-equilibrium effects, as key predictive variables. Training datasets are created from one-dimensional steady shock simulations, and a methodology of computing collision integrals is developed. By learning thermodynamically consistent closures directly from DSMC data, our R13-ML model, combined with a discontinuous Galerkin solver for the transfer equations of moments, preserves physical structure and accurately predicts normal shock structures and generalizes to hypersonic and some unsteady, one-dimensional wave scenarios. This work bridges machine learning with continuum mechanics, offering a road map for high-fidelity aerothermal predictions in next-generation supersonic vehicles.

physics.flu-dyn↗

arXiv2025

Fisher entropic Fokker-Planck model of monatomic rarefied gases

Particle-based stochastic approximations of the Boltzmann equation are popular tools for simulations of non-equilibrium gas flows, for which the Navier-Stokes-Fourier equations fail to provide accurate description. However, these numerical methods are computationally demanding, especially in the near-continuum regime, where the collisions become overwhelming. On the other hand, the Fokker-Planck kinetic models offer an efficient alternative, as the binary collisions are described by a diffusive process. Despite the intuitive advantage, rigorous and efficient Fokker-Planck approximations of the Boltzmann equation remain an open problem. On one hand, the moment projection of the Fokker-Planck operator should be consistent with that of the Boltzmann operator. On the other hand, the Fokker-Planck model should be constructed in such a way that the H-theorem is satisfied. The central aim of this study is fulfilling these two categorically different constraints, i.e. moment matching and entropy dissipation, within a flexible and tractable Fokker-Planck framework. To this end, we introduce a Fisher information-based entropic constraint and demonstrate that, with a suitable polynomial expansion of the drift term, it is possible to simultaneously achieve weak moment matching while honouring the H-theorem. We support our theoretical result by numerical experiments on the shock problem, validating our Fisher Entropic Fokker-Planck framework.

math.NA↗

arXiv2025

Paired Explicit Relaxation Runge-Kutta Methods: Entropy-Conservative and Entropy-Stable High-Order Optimized Multirate Time Integration

We present novel entropy-conservative and entropy-stable multirate Runge-Kutta methods based on Paired Explicit Runge-Kutta (P-ERK) schemes with relaxation for conservation laws and related systems of partial differential equations. Optimized schemes up to fourth-order of accuracy are derived and validated in terms of order of consistency, conservation of linear invariants, and entropy conservation/stability. We demonstrate the effectiveness of these P-ERRK methods when combined with a high-order, entropy-conservative/stable discontinuous Galerkin spectral element method on unstructured meshes. The Paired Explicit Relaxation Runge-Kutta methods(P-ERRK) are readily implemented for partitioned semidiscretizations arising from problems with equation-based scale separation such as non-uniform meshes. We highlight that the relaxation approach acts as a time-limiting technique which improves the nonlinear stability and thus robustness of the multirate schemes. The P-ERRK methods are applied to a range of problems, ranging from compressible Euler over compressible Navier-Stokes to the visco-resistive magnetohydrodynamics equations in two and three spatial dimensions. For each test case, we compare computational load and runtime to standalone relaxed Runge-Kutta methods which are outperformed by factors up to four. All results can be reproduced using a publicly available repository.

math.NA↗

arXiv2025

A finite element solver for a thermodynamically consistent electrolyte model

In this study, we present a finite element solver for a thermodynamically consistent electrolyte model that accurately captures multicomponent ionic transport by incorporating key physical phenomena such as steric effects, solvation, and pressure coupling. The model is rooted in the principles of non-equilibrium thermodynamics and strictly enforces mass conservation, charge neutrality, and entropy production. It extends beyond classical frameworks like the Nernst-Planck system by employing modified partial mass balances, the electrostatic Poisson equation, and a momentum balance expressed in terms of electrostatic potential, atomic fractions, and pressure, thereby enhancing numerical stability and physical consistency. Implemented using the FEniCSx platform, the solver efficiently handles one- and two-dimensional problems with varied boundary conditions and demonstrates excellent convergence behavior and robustness. Validation against benchmark problems confirms its improved physical fidelity, particularly in regimes characterized by high ionic concentrations and strong electrochemical gradients. Simulation results reveal critical electrolyte phenomena, including electric double layer formation, rectification behavior, and the effects of solvation number, Debye length, and compressibility. The solver's modular variational formulation facilitates its extension to complex electrochemical systems involving multiple ionic species with asymmetric valences. We publicly provide the documented and validated solver framework.

cs.CE↗

arXiv2025

A generalized fundamental solution technique for the regularized 13-moment system in rarefied gas flows

In this work, we explore the method of fundamental solutions (MFS) for solving the regularized 13-moment (R13) equations for rarefied monatomic gases. While previous applications of the MFS in rarefied gas flows relied on problem-specific fundamental solutions, we propose a generic approach that systematically computes the fundamental solutions for any linear moment system without predefined source terms. The generalized framework is first introduced using a simple example involving the Stokes equations, and is then extended to the R13 equations. The results obtained from the generic MFS are validated against an analytical solution for the R13 equations. Following validation, the framework is applied to the case of thermally-induced flow between two non-coaxial cylinders. Since no analytical solution exists for this case, we compare the results obtained from the MFS with those obtained from the finite element method (FEM). To further assess computational efficiency, we analyze the runtimes of the FEM and MFS. The results indicate that the MFS converges faster than the FEM and serves as a promising alternative to conventional meshing-based techniques.

physics.flu-dyn↗

arXiv2025

Sparse Reconstruction of Multi-Dimensional Kinetic Distributions

In the present work, we propose a novel method for reconstruction of multi-dimensional kinetic distributions, based on their representation as a mixture of Dirac delta functions. The representation is found as a solution of an optimization problem. Different target functionals are considered, with a focus on sparsity-promoting regularization terms. The proposed algorithm guarantees non-negativity of the distribution by construction, and avoids an exponential dependence of the computational cost on the dimensionality of the problem. Numerical comparisons with other classical methods for reconstruction of kinetic distributions are provided for model problems, and the role of the different parameters governing the optimization problem is studied.

physics.comp-ph↗

arXiv2025

Well-Posedness of the Linear Regularized 13-Moment Equations Using Tensor-Valued Korn Inequalities

In this paper, we finally prove the well-posedness of the linearized R13 moment model, which describes, e.g., rarefied gas flows. As an extension of the classical fluid equations, moment models are robust and have been frequently used, yet they are challenging to analyze due to their additional equations. By effectively grouping variables, we identify a 2-by-2 block structure, allowing us to analyze well-posedness within the abstract LBB framework for saddle point problems. Due to the unique tensorial structure of the equations, in addition to an interesting combination of tools from Stokes' and linear elasticity theory, we also need new coercivity estimates for tensor fields. These Korn-type inequalities are established by analyzing the symbol map of the symmetric and trace-free part of tensor derivative fields. Together with the corresponding right inverse of the tensorial divergence, we obtain the existence and uniqueness of weak solutions. This result also serves as the basis for future numerical analysis of corresponding discretization schemes.

math.AP↗

arXiv2024

Entropy-stable fluxes for high-order Discontinuous Galerkin simulations of high-enthalpy flows

In the present work, we extend the Discontinuous Galerkin Spectral Element Method (DGSEM) to high-enthalpy reacting gas flows with internal degrees of freedom. An entropy- and kinetic energy-preserving flux function is proposed which allows for use of arbitrary expressions for the internal energies of the constituent gas species. The developed method is applied to simulation of several model problems and compared to the DLR TAU solver.

physics.flu-dyn↗

arXiv2024

Entropy-conservative high-order methods for high-enthalpy gas flows

A framework for numerical evaluation of entropy-conservative volume fluxes in gas flows with internal energies is developed, for use with high-order discretization methods. The novelty of the approach lies in the ability to use arbitrary expressions for the internal degrees of freedom of the constituent gas species. The developed approach is implemented in an open-source discontinuous Galerkin code for solving hyperbolic equations. Numerical simulations are carried out for several model 2-D flows and the results are compared to those obtained with the finite volume-based solver DLR TAU.

physics.flu-dyn↗