This survey gives an overview of the Bernoulli or one-phase free boundary problem, with emphasis on recent progress.
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.
