Modern weather forecasting relies on the integration of observational systems, numerical modeling, data assimilation, high-performance computing, and increasingly artificial intelligence techniques. This paper reviews the scientific and technological foundations of contemporary weather prediction, with particular emphasis on the mathematical and computational tools that support operational forecasting. Weather prediction is based on the numerical solution of the nonlinear partial differential equations governing atmospheric dynamics. The accuracy of these forecasts critically depends on the availability of observations collected from heterogeneous sources, including ground-based networks, weather radars, satellites, aircraft, and other remote sensing platforms. These data are combined with model forecasts through advanced data assimilation techniques, which provide dynamically consistent estimates of the atmospheric state and improve forecast quality. Special attention is devoted to nowcasting, one of the most challenging areas of modern meteorology. At lead times of a few minutes to several hours, rapidly evolving phenomena such as severe thunderstorms, hailstorms, and flash floods require the integration of high-frequency observations, radar extrapolation methods, ensemble prediction techniques, and machine learning approaches. Artificial intelligence is increasingly used to enhance the detection, tracking, and short-term prediction of convective systems, complementing traditional physics-based methodologies. The paper also discusses the requirements of a modern national forecasting system and presents the high-resolution ICON-2I modeling framework operational at the ItaliaMeteo Agency. Particular emphasis is placed on the role of high-performance computing infrastructures, ensemble forecasting systems, and kilometer-scale data assimilation methods. Finally, current challenges and future developments are examined, including convective-scale prediction, uncertainty quantification, exascale computing, digital twins of the atmosphere, and the growing integration of artificial intelligence with numerical weather prediction.
This paper considers a model describing population dynamics where diffusion is nonlocal and habitat boundaries expand based on population flux. The population distribution is governed by a nonlocal Fisher–KPP equation with free boundaries. We propose a Front-Tracking method coupled with a fourth-order Runge–Kutta scheme for time integration, which adaptively manages the computational grid. For comparison, we outline the Front-Fixing approach, which transforms the problem into a fixed domain. Numerical simulations are performed for different scenarios, comparing the accuracy and behavior of both proposed methods. Furthermore, a variance-based global sensitivity analysis using Sobol’ indices is conducted to identify the influence of key model parameters on the final habitat length. Results show that the initial habitat size and the boundary expansion rate are the most influential parameters affecting the final domain length, with their relative importance depending on the distributions of the chosen input parameters.
Inertial Confinement Fusion energy is one of the most promising technologies for the future energy transition, yet several challenges must be addressed before a fully operational reactor can be realized. Among the most relevant open issues is the design of the reaction vessel, including material selection, geometric configuration and functional characteristics. Computational frameworks offer a powerful means to explore the design space, and several examples already exist in the context of magnetic confinement fusion. This work reviews the main physics-based computational tools used to investigate Inertial Confinement Fusion systems, including dedicated codes such as BUCKY-1 and CHIC, as well as more general-purpose solvers for thermal and mechanical stress analysis commonly adopted to study fusion-related problems arising from plasma generation and containment. In addition, the paper introduces recent developments in machine learning for design optimization, with particular attention to Multi-Fidelity Bayesian Optimization Methods, which have been successfully applied to inertial fusion targets and represent a promising approach for guiding future vessel design.
The paper presents an hp analysis of Nitsche’s method applied to Maxwell’s problem using a stabilised finite element (FE) formulation. While the FE formulation and the weak imposition of Dirichlet boundary conditions have been previously explored, the novelty of this work lies in the hp mesh refinement analysis, examining the method’s behaviour as both the mesh size h decreases and the polynomial order p increases. The study specifically aims to adapt previous analyses to account for these refinements and to highlight a slight lack of optimality in p that is intrinsic to Nitsche’s method.
Quantum coherence, once mainly studied in atomic and optical systems, now offers potential for energy conversion technologies. It influences light absorption and emission, affecting energy conversion limits and efficiency. As a result, quantum coherence is being harnessed to boost performance in quantum heat engines, photocells, and photosynthetic-inspired platforms. Of particular interest in this context is the generation of Fano coherences, i.e., the formation of quantum coherences due to the interaction with the continuum of modes characterising an incoherent process. We aim to formalize mathematically the possibility of achieving steady-state Fano coherence in a V-type three-level quantum system using polarized incoherent radiation, without requiring the energy difference between the excited levels, ħΔ , to tend to zero. Specifically, in this scenario, it can be shown that the maximum steady-state Fano coherence is obtained through a non-trivial interplay between Δ , the average spontaneous decay rate γ̅ of the excited levels, and the intensity of the incoherent radiation, quantified by the mean photon number n̅ . We perform this analysis by deriving the Bloch–Redfield equation from first-principles by quantizing the incoherent radiation. The resulting reduced dynamics of the system are analysed, so as to determine the lifetime of Fano coherence and identify the conditions under which it becomes stationary. We characterise distinct dynamical regimes, ranging from weak to strong pumping, in which steady-state Fano coherence emerges, and we quantitatively determine its magnitude. For each regime, we analyse the generation of Fano coherence as a function of both the intensity of the incoherent pumping and the energy splitting between the excited levels. We also assess how obtaining Fano coherence is modified by symmetric or asymmetric decay rates. These findings indicate that a three-level quantum system driven by polarized incoherent light can act as a robust resource for coherence-assisted energy conversion and storage. Finally, we discuss the experimental challenges associated with the implementation of the proposed model using an ensemble of Rubidium atoms. This work paves the way towards the forthcoming realization of experimental tests of Fano coherence generation in a relevant experimental setup by making use of a polarized incoherent radiation, in order to achieve quantum-enhanced energy storage and energy-conversion functionalities.
Physics-based models play a central role in the quantitative understanding and simulation of lithium–ion batteries. Among them, the Doyle–Fuller–Newman (which we will refer to as DFN) model, also known as the pseudo-two-dimensional (P2D) model, has become the reference continuum framework for describing the coupled transport, electrochemical, and kinetic phenomena governing cell operation. While widely used in both academia and industry, the mathematical structure of the DFN model, its underlying assumptions, and the numerical strategies employed for its solution are often presented in a fragmented or application-driven manner. This review provides a coherent and mathematically oriented overview of the DFN framework, with particular emphasis on its multiscale formulation, governing equations, and numerical implementation. We first summarise the physical and electrochemical principles underlying lithium–ion cell operation and show how these mechanisms are translated into the DFN model through porous-electrode theory. We then present the resulting coupled system of partial differential equations, highlighting its multiscale structure and nonlinear couplings. A dedicated part of the review is devoted to numerical solvers and implementations for DFN-type models. We discuss representative finite-element and finite-volume approaches, including COMSOL Multiphysics, DandeLiion, and PyBaMM, and analyse how solver design choices influence robustness, accuracy, and behaviour near limiting regimes. A comparative numerical study illustrates how different treatments of stiffness and constraints can lead to solver-dependent predictions close to end-of-discharge. By emphasising the interplay between physical modelling assumptions and their mathematical and numerical consequences, this work provides applied mathematicians and numerical analysts with a unified perspective on DFN-based battery modelling.
We study a mathematical model of bacterial growth on a single limiting nutrient in a chemostat where a virus is present. The assumption is that the virus can infect the population, resulting in the emergence of two distinct populations: susceptible and infected, which are in competition. The model has the structure of an SIS epidemic model. We assume that the growth functions are general and not just linear as in previous studies in the literature. We analyze the stability of both disease-free and endemic equilibria. The model can exhibit a multiplicity of endemic equilibria, as well as the appearance of periodic orbits by supercritical or subcritical Hopf bifurcations. Bistability between several equilibrium states or limit cycles is also possible. We present an explicit expression for the basic reproduction number of the epidemic in terms of biologically significant parameters. To better understand the richness of the model’s behavior, some bifurcation diagrams are examined with respect to input nutrient concentration.
This paper provides a broader perspective of the extended Lorenz-84 and the high-order Lorenz–Stenflo systems. This perspective allows influential parameters to vary with time. Based on these time-varying parameters, we derive sufficient conditions for several global dynamical properties, including attractivity under vanishing forcing terms, as well as asymptotic and (uniform) exponential stability of the systems in their unforced forms. These ubiquitous features are analyzed and discussed by numerical simulations.
Modern computer hardware performs reduced-precision floating-point computations orders of magnitude faster than double precision. Yet these speedups cannot readily be exploited for scientific computing since many numerical algorithms lack robustness to rounding error accumulation at such low precisions. This review article focuses on numerical timestepping methods and surveys theory and strategies for reduced- and mixed-precision Runge–Kutta methods—respectively performing computations entirely or selectively in low precision formats. We show that rounding error analysis can guide the design of algorithms that produce sufficiently accurate numerical results at possibly a fraction of the computational cost. In particular, the rounding error behaviour of time-stepping schemes is closely tied to error accumulation in recursive summation and algorithmic techniques developed to mitigate rounding error growth in summations can be adapted to time integration. Three key strategies are relevant to timestepping and are discussed here: compensated summation, unbiased stochastic rounding, and mixed precision. Whereas Naïve reduced-precision implementations cause rounding errors to blow-up as the timestep decreases, we describe reduced-precision techniques that mitigate, or even eliminate, rounding error growth. Moreover, we present mixed-precision Runge–Kutta strategies that can additionally recover the full convergence order of the scheme as if run in exact arithmetic. The article concludes with an outlook on open challenges and future research directions in reduced- and mixed-precision time integration.
Hydrogen has long been considered a potential cornerstone of the future energy system, especially in the context of the transition away from fossil fuels. Its appeal lies in its abundance, its clean combustion (producing only water as a by-product), and its versatility as both a fuel and a feedstock. However, the practical use of hydrogen as an energy vector is hindered by fundamental challenges related to its physical and chemical properties. Its very low atomic mass makes hydrogen an energy carrier with low volumetric density; liquefaction is energetically costly due to the extremely low boiling point, and compression requires pressures of several hundred atmospheres, raising issues of safety and efficiency. Furthermore, hydrogen transport is problematic: its tiny molecular size leads to permeation through pipes, and it can induce embrittlement and cracking in metals such as steel. Alternative strategies, such as binding hydrogen into compounds like ammonia, provide partial solutions but bring their own limitations, including energy-intensive synthesis. A promising avenue of research involves solid-state hydrogen storage using metals or alloys, notably palladium, which exhibits exceptional absorption properties. The aim of this article is to discuss these issues from a mathematical and physical perspective, highlighting the analytical and modeling challenges that arise when hydrogen is considered as a key element of the energy transition .
This study presents a Mean Value Theorem (MVT) based convergence analysis for a highly efficient iterative algorithm of order eight studied by Cordero et al. (2018) for solving nonlinear systems of equations. Using the same idea, one can increase the order of convergence (OC) from p to p+3 for any algorithm with OC p≥ 5. The algorithm involves a weight function, so one can introduce new algorithms by choosing the weight function appropriately. We have proved the OC using MVT and in a Banach space setting. Both local and semi-local analysis are discussed under the same set of assumptions. Our study used the derivative of T only up to the third order. This algorithm can be applied to solve complex problems from various scientific disciplines. Algorithm’s performance is illustrated using numerical examples.
This work aims to investigate the exponential stability of a one-dimensional thermoelastic laminated beam with structural damping, considering the effect of the heat flux governed by the Coleman–Gurtin thermal law on the effective rotation angle equation. Using the energy method, we demonstrate the exponential stability if the wave speeds are equal. Furthermore, using Gearhart–Herbst–Prüss–Huang’s theorem, we demonstrate the lack of exponential stability if the wave speeds are different.
In this paper, we consider the structured model updating problem for matrix pencils that arise in network systems. The main objective of this paper is to determine structure-preserving perturbations of matrix pencils associated to network systems in such a manner that a perturbed matrix pencil generates a specified set of eigenvalues while substituting a collection of given eigenvalues of the unperturbed pencil and preserving the corresponding eigenvectors, which is referred to as the model updating problem in the context of structural dynamics. We determine a parametric solution of the above problem. Finally, the results are illustrated with the help of numerical examples.
We investigate the application of optimal control theory to waste management procedures, in particular to anaerobic degradation processes in bioreactor systems, to highlight how mathematical tools can inform decision-makers. Starting from a mathematical model describing the interactions among soluble substrate, insoluble substrate, and biomass, we formulate and analyze both finite-horizon and time optimal control problems in which leachate recirculation acts as the control variable. In the finite-horizon framework, optimal strategies are designed to balance substrate reduction and operational costs. In contrast, the time-optimal formulation aims to minimize the time required for the system to reach a prescribed target. Analytical insights derived from the application of Pontryagin Minimum Principle, complemented by numerical simulations, allow for the characterization of the optimal control and the analysis of the role played by the initial substrate composition, operational costs, and target shape on the resulting strategies.
We present an efficient method for solving a class of 4× 4 block saddle-point systems arising from the finite element discretization of the generalized three-dimensional Stokes problem. The spectral properties of the preconditioned system are investigated, including the distribution of eigenvalues and the behavior of the associated eigenvectors. To efficiently handle multiple right-hand sides within the resulting subsystems, we propose the Preconditioned Global Conjugate Gradient (PGCG) method as a block iterative solver and establish new convergence results. Numerical experiments demonstrate that the proposed preconditioned iterative approach substantially improves the efficiency of solving the 3D Stokes problem.
We study a stochastic optimal control problem motivated by the operation of a large ensemble of residential storage devices coordinated by an energy aggregator. The aggregator remunerates prosumers in exchange for direct control of their batteries and seeks to jointly (i) reduce local supply–demand imbalances and (ii) exploit intraday price fluctuations through energy arbitrage. The core modeling feature is a kinetic mean-field formulation: the state of charge is treated as a position, the charging/discharging power as a velocity, and the control as an acceleration, thus encoding ramp-rate limitations and producing smooth power trajectories. This leads to a controlled McKean–Vlasov Langevin-type system in which both the drift and the objective functional depend on the time-marginal law of the state, allowing one to capture endogenous interaction effects and population-level stabilization incentives. The performance criterion combines the cost of grid exchange with convex penalties representing degradation and control effort, and includes mean-field terms that promote alignment with the population average; terminal contributions account for residual energy value and end-of-horizon coordination. The resulting control problem is Markovian and hypoelliptic, and naturally connects mean-field control with ultraparabolic operators of kinetic type. This viewpoint provides a coherent bridge between physically constrained storage actuation and law-dependent incentives in large-scale energy management. Numerical experiments based on deep learning solvers are presented to validate the model. From a computational standpoint, the problem is particularly challenging, as it yields a fully coupled forward-backward stochastic system associated with a five-dimensional Hamilton–Jacobi–Bellman equation.
In two well-known studies [Mathematics and Computers in Simulation 79(2008) 622–633] and [Mathematics and Computers in Simulation 182(2021) 397–410], nonstandard finite difference (NSFD) schemes for an SIRC epidemic model of influenza A have been proposed. There have been attempts to prove that these NSFD schemes preserve essential dynamical properties of the continuous-time model for all finite step sizes. These properties include the positivity of the solutions, the invariance of the total population (conservation law), equilibrium points and their asymptotic stability. Although the SIRC model possesses a unique disease-free equilibrium (DFE) point and a unique disease-endemic equilibrium (DEE) point, only the local asymptotic stability (LAS) of the DFE point has been established theoretically, whereas the LAS of the DEE point has only been confirmed through numerical simulations for specific parameter sets. In this work, we construct a new class of NSFD schemes for the SIRC epidemic model. The LAS of the equilibrium points of the constructed NSFD schemes is rigorously established from a theoretical perspective and validated through numerical experiments. These NSFD schemes are constructed based on a weighted approximation for linear terms and the renormalization of the denominator function. Thereafter, we derive dynamic consistency thresholds that lead to easily-verified conditions, ensuring the NSFD schemes preserve all the essential dynamical properties of the continuous-time model, regardless of the step size. In particular, the LAS of the constructed NSFD schemes can be easily established based on the linearized method thanks to their simple structure. Furthermore, they produce numerical approximations with smaller errors than the existing NSFD schemes for a given parameter set. Additionally, Richardson’s extrapolation technique can be conveniently applied to increase the accuracy of the constructed NSFD schemes. Consequently, we obtain a new class of dynamically consistent NSFD schemes, which is simple and efficient for numerical simulation of the SIRC model. Finally, numerical experiments are performed to support the theoretical findings and to demonstrate the advantages of the proposed NSFD schemes.
This work presents a fractional-order mathematical model featuring a generalized incidence function, designed to encapsulate the inherent memory effects observed in biological systems. The model incorporates partial vaccination coverage and accounts for complex transmission pathways, thereby elucidating critical yet often neglected mechanisms that are essential for understanding and managing pathogen propagation effectively. A mathematical analysis is conducted to validate the biological soundness of the model and to characterize the asymptotic behavior of its solutions. The disease-free equilibrium is globally stable if ℛ_0≤ 1 , while a unique, globally stable endemic equilibrium exists if ℛ_0> 1 , as rigorously proven via Lyapunov functions and graph theory. A sensitivity analysis ranks key parameters to guide public health interventions. Finally, the operational relevance of the model is illustrated and validated through numerical simulations calibrated with historical cholera outbreak data, confirming its utility for epidemic forecasting and the design of effective containment strategies.
In this paper, we formulated a mathematical model for the transmission dynamics of corruption by explicitly considering the behavior of the Criminal Justice System. The model was proven to be mathematically well posed. We used the next-generation matrix method to calculate the basic reproduction number regarding the corruption-free equilibrium and established the conditions for the local and global asymptotic stability of these equilibrium points. The analysis reveals a globally asymptotically stable corrupt-free equilibrium whenever ℛ<1 and a globally asymptotically stable endemic equilibrium if otherwise. The sensitivity analysis and numerical simulations showed that integrating a strategy aiming to strengthen the law and criminal justice system, as well as the campaigns to raise the awareness against corruption is the best approach to fight this phenomenon.
A numerical integrator for ẋ=f(x) is called stable if, when applied to the 1D Dahlquist test equation ẋ=λ x,λ∈ℂ with fixed timestep h>0 , the numerical solution remains bounded as the number of steps tends to infinity. It is well known that no explicit integrator may remain stable beyond certain limits, depending on the domain of λ . Furthermore, these stability limits are only tight for certain specific integrators (different for each domain), which may then be called ‘optimally stable’. Such optimal stability results are typically proven using sophisticated techniques from complex analysis, leading to rather abstruse proofs. In this article, we pursue an alternative approach, exploiting connections with the Bernstein and Markov brothers inequalities for polynomials. This simplifies the proofs greatly and, moreover, offers a simple framework which unifies the diverse results that have been obtained.
