Moment flux-based discrete velocity method based on the Boltzmann-BGK equations along the characteristic line

Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient methods to the system class of nonlinear port-Hamiltonian differential-algebraic equations - as they emerge from the port- and energy-based modeling of physical systems in various domains. We introduce a novel numerical scheme tailored for semi-explicit differential-algebraic equations and further address more general settings using the concepts of discrete gradient pairs and Dirac-dissipative structures. Additionally, the behavior under system transformations is investigated and we demonstrate that under suitable assumptions port-Hamiltonian differential-algebraic equations admit a representation which consists of a parametrized port-Hamiltonian semi-explicit system and an unstructured equation. Finally, we present the application to multibody system dynamics and discuss numerical results to demonstrate the capabilities of our approach.

• Deterministic and stochastic methods are studied for adjoint-based inverse modeling. • Novel stochastic update rule preserves boundary and regularity via adjoint-state. • When coupled with high-order schemes, it enhances smoothness of tuning parameter. • New method outperforms L-BFGS and N-ADAM in multi-objective inverse problems. This work explores the use of quasi-Newton and stochastic optimization methods for gradient-based inverse problems in physical modeling, focusing on their application within high-order numerical frameworks. Such problems are often characterized by the following challenges: (i) physical constraints often yield a highly non-convex design space with multiple local optima; (ii) solutions must satisfy intrinsic properties, such as boundary and regularity conditions, which are not easily enforced as explicit constraints. Conventional stochastic optimizers, such as N-ADAM, exhibit significant information loss in their update rules, leading to non-physical solutions. To overcome these limitations, we introduce a novel stochastic optimizer, V-N-ADAM-DG, which incorporates adjoint-state information into the update rule to maintain physically meaningful corrections in terms of regularity and boundary conditions. We validate our approach in the context of mean-flow reconstruction for Reynolds-averaged Navier-Stokes (RANS) simulations using a high-order discontinuous Galerkin (DG) discretization method, as proposed by Fanizza et al. (2025). The optimization framework considers both vectorial corrective terms, inferred in the momentum and energy equations, and scalar corrective terms in the Spalart-Allmaras (SA) transport equation. The V-N-ADAM-DG optimizer effectively reconstructs mean flow quantities while ensuring smooth transitions of corrective parameters at boundaries, an improvement over standard stochastic optimizers. Additionally, it facilitates a rapid decay of the optimal degrees of freedom (DOFs), leading to smoother corrections in high-order reconstructions-achieving a balance between the robustness of quasi-Newton methods (such as L-BFGS) and the flexibility of stochastic approaches. Numerical experiments across various flow configurations demonstrate that V-N-ADAM-DG consistently outperforms both L-BFGS and N-ADAM, particularly in complex inverse problems that employ multiple combined cost functions to reconstruct different physical quantities.

Novel discretization method to calculate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si343.svg" display="inline" id="d1e3906"> <mml:mi>g</mml:mi> </mml:math> -functions of vertical geothermal boreholes with improved accuracy and efficiency

Optimal L2 error estimates for a fully discrete method of the Cahn-Hilliard-MHD model

High-fidelity neutronic analyses of advanced reactors require deterministic transport solvers capable of handling complex unstructured geometries while maintaining computational efficiency. This work presents the development and verification of three graphics processing unit (GPU)–accelerated deterministic solvers implemented within a unified framework, Neutronics using Deterministic Finite Element Algorithm (NuDEAL): the planar method of characteristics (MOC) coupled with the hybrid finite element method (HFEM), the discontinuous Galerkin method of characteristics (DGMOC), and the discontinuous finite element discrete ordinate method (DFEM-SN). These solvers provide complementary capabilities for consistently solving the multigroup transport equation and can be selectively employed to balance accuracy, computational cost, and memory requirements for a given problem. All the methods emphasize efficient GPU execution by leveraging memory alignment, compressed flux storage, and sequential azimuthal sweeps. The solvers are validated on the C5G7 benchmark and applied to advanced reactor problems, including the Advanced Burner Test Reactor (ABTR), Empire microreactor, and the Molten Salt Reactor Experiment. DFEM-SN achieved the highest accuracy, with eigenvalue errors below 50 pcm, while MOC/HFEM and DGMOC provided superior efficiency, with single-GPU run times comparable to those of large CPU clusters. The results demonstrate that deterministic GPU solvers on unstructured meshes can deliver both accuracy and scalability, enabling practical whole-core simulations for heterogeneous advanced reactors. The unified NuDEAL framework establishes a foundation for future extensions toward transient and multiphysics analyses on large-scale GPU architectures.

This study introduces a discrete element numerical simulation method for calcareous sand particles that incorporates a dual-porosity and allows for the quantitative generation of internal pores. The method is designed to capture three key distinguishing characteristics of calcareous sand compared to quartz sand: the abundant dual-porosity, pronounced particle aspect ratio and high susceptibility to breakage. Biaxial tests are simulated across various confining pressures to explore the effects of different particle fragmentation states, as well as the impact of internal and external pores on the shear strength of the sample. The findings of this study offer significant academic insights into the strength and fragmentation mechanisms of calcareous sand, thus contributing to the broader understanding of granular materials in geotechnical engineering and related fields.

Abstract We propose a dual-resolution prescription for the Boltzmann neutrino transport, in which the advection and collision terms are calculated at different angular resolutions in momentum space. The purpose is to address the issue of the low resolution that afflicts the discrete ordinates method in multidimensional neutrino transport simulations of core-collapse supernovae. We handle the advection term at a high resolution, assuming that the collision term does not require such high resolution. To confirm this surmise as well as our new conversion scheme, from low-to-high angular resolutions and vice versa, we run a couple of experimental one-zone (in space) simulations. Neutrino scattering on nucleons is considered with small recoils fully taken into account, whereas the advection term is replaced by the angle- and energy-dependent source terms that are designed to mimic the results of a Boltzmann simulation, inducing anisotropy in momentum space. For the conversion from a low-resolution distribution function to a high-resolution one, we employ polynomial interpolations in the zenith ( μ ν ) and azimuth ( ϕ ν ) directions separately, with number conservation and continuity (and periodicity only in the ϕ ν direction) imposed. We find that this dual-resolution scheme works well and that the current angular resolution employed in the canonical supernova simulations with our Boltzmann solver, or a bit better in the ϕ ν direction, will be sufficient for the collision terms if they are coupled with the advection terms calculated with a high angular resolution via this prescription.

Abstract We analyze an advection–diffusion–reaction problem with non-homogeneous boundary conditions that models the chromatography process. We prove stability and error estimates for both constant and affine adsorption, using the symplectic one-step implicit midpoint method for time discretization and finite elements for spatial discretization. In addition, we perform the stability analysis for the nonlinear, explicit adsorption in the continuous and semi-discrete cases. For the nonlinear, explicit adsorption, we also complete the error analysis for the semi-discrete case and prove the existence of a solution for the fully discrete case. The numerical tests validate our theoretical results.

This paper investigates the feasibility of using Large Language Models (LLMs) for the structural analysis of flow-based network data. This analysis is carried out in the presence of a structural difference between the multidimensional numerical space of IoT features and the symbolic space in which LLMs operate. The primary objective was the development of a formal framework that enables the controlled transformation of numerical data into linguistically analyzable semantic representations, without resorting to classification or machine learning mechanisms. We propose the Semantic Flow Encoding (SFE) mechanism, a deterministic method for robust discretization and behavioral abstraction that converts the numerical characteristics of Internet of Things (IoT) flows into structural semantic descriptions using the Canadian Institute for Cybersecurity Internet of Things Device Identification and Anomaly Detection (CIC IoT-DIAD) 2024 dataset. Through formal informational measures, it is demonstrated that the existence of an intrinsic structural difference between benign and DDoS traffic in the analyzed dataset. In the validation stage, we evaluated whether these informational differences are reflected at the level of linguistic abstraction through controlled inference experiments in IBM WatsonX. The present paper suggests that LLMs may support semantic auditing of distributional structure when guided by a formal encoding layer. In this manner, a reproducible framework for integrating numerical security data into language-model-based analysis is suggested.

A discrete–time model of co–infection dynamics in a heterogeneous population is presented. In such a population we distinguish two subpopulations with different levels of individual susceptibility to any of the two diseases. As a discretization method we use a non–standard discretization scheme that preserves positivity of system's variables. The novelty of our works lies in assuming that each parameter describing a specific biological process has a different value for each subpopulation. This assumption makes the investigated population completely heterogeneous. We investigate a local stability analysis of a disease–free stationary state. Because of our system's complexity, we only give insight into local stability of equilibria that correspond to the presence of a single disease. For such states we provide conditions for their local stability, considering boundary cases. We conduct numerical simulations that suggest such postulated stability. Moreover, the simulations show that there should exist a locally stable state, which reflects the simultaneous presence of both diseases. The simulations also illustrate the concept of an infection rate, which helps compare the impact of both diseases on the dynamics of co–infection spread.

Algebraic Multigrid (AMG) methods are state-of-the-art algebraic solvers for Partial Differential Equations. Still, their efficiency depends heavily on the choice of suitable parameters and/or ingredients. Paradigmatic examples include the so-called strong threshold parameter, which controls the algebraic coarse-grid hierarchy, as well as the smoother, i.e. the relaxation methods used on the fine grid to damp out high-frequency components of the error. In AMG, since the coarse grids are constructed algebraically (without geometric intuition), the smoother’s performance is even more critical. For the linear systems stemming from polytopal discretizations, such as Polytopal Discontinuous Galerkin (PolyDG) and Virtual Element (VEM) methods, AMG sensitivity to such choices is even more critical due to the significant variability of the underlying meshes, which results in algebraic systems with different sparsity patterns. In this paper, we focus on the linear systems of equations stemming from polytopal discretizations of second-order elliptic problems. We propose a novel deep learning approach that automatically tunes the strong threshold parameter and the smoother choice in AMG solvers, thereby maximizing AMG performance. We test various differential problems in both two- and three-dimensional settings, with heterogeneous coefficients and polygonal/polyhedral meshes, and demonstrate that the proposed approach generalizes well. In practice, we demonstrate that we can reduce AMG solver time by up to [Formula: see text] with minimal changes to existing PolyDG and VEM software libraries.

The discrete ordinates method has served as a cornerstone of numerical radiative transfer since A. Schuster introduced its foundation in 1905. G.C. Wick and S. Chandrasekhar significantly advanced the method in the mid-20th century to study planetary atmospheres. In 1963, B.G. Carlson formally applied the method to neutron transport, initiating its widespread use thereafter. Since then, the discrete ordinates method has grown from simple linear interpolation to sophisticated multidimensional algorithms applied to reactor analysis and weapons design. In this work, we consider a 1D, monoenergetic response matrix discrete ordinates method to solve the even-parity, second-order form analytically with hyperbolic matrix functions. Our focus then turns to incorporating a nascent delta function source (DFS) through faux interpolation of discrete angular fluxes. Finally, we benchmark the DFS approach against the conventional first collision source (FCS) to demonstrate agreement to eight or nine significant digits.

The multifracture competitive initiation under different fracturing modes is of great significance for designing hydraulic fracturing treatments. In this paper, a numerical model of shale oil reservoir fracturing considering fluid-mechanics coupling is established based on a three-dimensional (3D) discrete lattice method, and the properties of shale are characterized by the combination of rock matrix and weak mechanical layer. In addition to comparing the formation mechanism of complex fractures under conventionally oriented perforation and spiral perforation, radial well fracturing is also specifically considered. Furthermore, taking spiral perforation as an example, the influences of perforation phase, diameter, and density on near-wellbore fracture propagation are further analyzed. The results show that radial well fracturing can significantly reduce fracture initiation pressure and obtain a larger fracture area than perforation fracturing. For the perforation fracturing, the near-wellbore fracture morphology of oriented perforation is single, mainly forming multiple parallel planar fractures. Near-wellbore fracture morphologies of spiral perforation are complex and varied, including single planar fractures, single spiral fractures, double spiral fractures, and stepped spiral fractures. Appropriately decreasing the perforation phase and increasing the perforation diameter and density can reduce the number of failed perforations and promote the interconnectivity of fractures, thus forming the main fracture that communicates multiple perforation tunnels. The accuracy of the model is further verified by laboratory experiments under the same conditions as those of the numerical simulation. The results can provide theoretical guidance for the optimization of the fracturing parameters in shale oil reservoirs.

We simulate the dynamics of pelagic Sargassum rafts as systems of finite-size floating particles, governed by a Maxey-Riley law with nonlinear elastic interactions. Using surface ocean currents and wind data from reanalysis systems for clump transport, we computed trajectories within a domain covering the tropical and subtropical north Atlantic. The subsequent motion is reduced using Ulam's discretization method into a time-inhomogeneous Markov chain that simulates a background Sargassum concentration. Bayesian inversion, combined with nonautonomous transition path theory, was used to infer the origin of the first significant recorded bloom in the tropical North Atlantic, which unfolded in April 2011. Both methodologies independently identified the bloom's origin as near the West African coast, up to 2 years before it was detectable via satellite imagery on the basin's western side. This finding supports anecdotal evidence of Sargassum strandings on the Ghanaian coast in 2009. Moreover, it correlates with unusual environmental conditions-such as increased nutrient loads from significant upwelling linked to a pronounced Dakar Niña and Saharan dust deposition-that promote bloom proliferation. Additionally, it aligns with the observation that the species of Sargassum in the 2011 bloom differ from those in the Sargasso Sea, which might otherwise be considered a natural origin.

Two absolute-value based preconditioned MINRES methods for Crank-Nicolson discretization of the viscoelastic equation

This paper presents a phase-field model for simulating the three-dimensional deformation of vesicle membranes, incorporating area-difference elasticity (i.e., the elasticity arising from the difference between the inner and outer lipid leaflets), with constraints on bulk volume and surface area. We develop efficient numerical schemes based on the Fourier-spectral method for spatial discretization and temporal evolution. The model successfully captures a wide variety of steady-state vesicle shapes. The numerical experiments demonstrate that by tuning the simulation parameters, the vesicle can transition from a simple spherical and discocyte shape to complete membrane fission, asymmetric pear shaped structures, as well as complex multi-armed starfish-like and nested configuration. These results highlight the crucial role of area-difference elasticity in determining vesicle morphology.

This paper develops a discrete coupled-mode method for wave propagation in a cylindrically symmetric fluid-solid medium with a symmetric point source on the vertical symmetry axis. Range dependence is handled by a discretization of the medium into laterally homogeneous ring regions. Modal reflection (or scattering) matrices, recursively computed by an initial inward propagation step using mode orthogonality, relate the expansion coefficients for outgoing and incoming (back-scattered) normal modes. Incorporation of the source condition yields the field in the innermost ring region. A subsequent outward propagation step, stabilized by the stored reflection matrices to satisfy the outer boundary condition, yields the modal expansion coefficients of the field in each ring region. With fluid overlying a solid bottom, upsloping and downsloping parts of the medium require different propagation equations to satisfy the continuity conditions at the vertical ring-region interfaces. An approximate solution appears by replacing the involved Hankel functions by their asymptotic expressions for large arguments. Compared to the solution for a symmetric horizontal line source in a corresponding medium, invariant in the line-source direction, the approximate solution involves a back-scattering term as well as a mode-dependent and phase-changing source-correction factor in addition to the correction factor for the different geometrical spreading.

The traveling salesman problem (TSP) is a classic combinatorial optimization problem belonging to the NP-hard problem. This paper extends the QUATRE algorithm to this field. The quasi-affine transformation evolution (QUATRE) algorithm provides six evolution schemes and simplifies the setting of control parameters. Because the QUATRE algorithm is easy to fall into local optimal and the convergence performance of the QUATRE algorithm is insufficient for the application, many excellent heuristic algorithms are continuously proposed. This article modified the QUATRE algorithm using reverse learning and mutation strategy. In order to expand the application of the QUATRE algorithm to the TSP problem, a discretization method is adopted to improve the QUATRE. It proposes the discrete parallel quasi-affine transformation evolution (DPQUATRE) algorithm. DPQUATRE beats the comparison algorithm on all 14 test sets of the Traveling Salesman problem library (TSPLIB). TSPLIB is utilized to assess the property of the DPQUATRE algorithm to show the effectiveness of the method. In addition, the error rate metrics PDBest and PDAverage are also used to evaluate the performance of the algorithms. The error rate provides a more visual demonstration of the gap between the distance calculated by the algorithm and the shortest distance.

ABSTRACT This paper investigates a modification of the fictitious domain method with continuation in the lower‐order coefficients for the unsteady Navier‐Stokes equations governing the motion of an incompressible homogeneous fluid in a bounded 2D or 3D domain. The modification enables a solution‐dependent choice of the critical parameter. Global‐in‐time existence and convergence of a weak solution to the auxiliary problem are proved, and local‐in‐time existence and convergence of a unique strong solution are established. For the strong solution, a new higher‐order convergence rate estimate in the penalization parameter is obtained. The introduced framework allows us to apply a pointwise divergence‐free finite element method as a discretization technique, leading to a strongly mass conservative discrete fictitious domain method. A numerical example illustrates the performance of the method.