Introduction. We present a novel adaptive mesh refinement (AMR) solver, SWqgdAMR, based on the open software platform AMReX. The new solver is grounded in regularized shallow water equations. This paper details the equations, their discretization, and implementation features within AMReX. The efficacy of SWqgdAMR is demonstrated through two test cases: a two-dimensional circular dam break (collapse of a liquid column) and the collapse of two liquid columns of different heights.
Materials and Methods. The SWqgdAMR solver is developed to extend the applicability of regularized equations in problems requiring high computational power and adaptive grids. SWqgdAMR is the first solver based on the quasigas dynamic (QGD) algorithm within the AMReX framework. The implementation and validation of SWqgdAMR represent a crucial step towards the further expansion of the QGD software suite.
Results. The AMReX-based shallow water equations solver SWqgdAMR with adaptive mesh refinement is described and tested in detail. Validation of SWqgdAMR involved two-dimensional problems: the breach of a cylindrical dam and the breach of two cylindrical dams of different heights. The presented solver demonstrated high efficiency, with the use of adaptive mesh refinement technology accelerating the computation by 56 times compared to a stationary grid calculation.
Discussion and Conclusions. The algorithm can be expanded to include bathymetry, external forces (wind force, bottom friction, Coriolis forces), and the mobility of the shoreline during wetting and drying phases, as has been done in individual codes for regularized shallow water equations (RSWE). The current implementation of the QGD algorithm did not test the potential for parallel computing on graphical cores.

Introduction. The study of heat interaction processes and the distribution of heat flaxes in the oceans is important for understanding climate change on Earth. The North Atlantic, which is one of the key components of the global climate system, plays a significant role in regulating the climate of our latitudes. One of the key tools for analyzing heat distribution in the oceans is probabilistic analysis. In this work, using mathematical modelling methods, a statistical analysis of observational data on heat fluxes in the North Atlantic is carried out.
Materials and Methods. The used methods include the analysis of random processes specified by the stochastic differential equation (SDE) or the Ito equation, approximation of observational data, and solution of the Fokker-Planck-Kolmogorov (FPK) equation to describe the evolution of the probabilistic distribution of heat in the ocean.
Results. Using mathematical modelling methods, a probabilistic analysis of the distribution of heat fluxes in the North Atlantic for the period from 1979 to 2022 has been carried out. The results of the study made it possible to establish patterns of distribution of heat flux in the studied region over the period of time under consideration.
Discussion and Conclusions. The results may be useful for further study of climate processes in the North Atlantic, as well as for the development of resource management and environmental protection strategies.

Introduction. This work is devoted to the mathematical modelling of extreme sea level fluctuations in the Azov Sea using remote sensing data. The aim of the study is to develop and apply a mathematical model that allows more accurate prediction of surge and seiche events caused by extreme wind conditions. The relevance of the work is due to the need to improve the forecasts of hydrodynamic processes in shallow water bodies (such as the Azov Sea), where such phenomena can have significant economic and ecological consequences. The goal of this work is to develop and apply a mathematical model for predicting extreme sea level fluctuations in the Azov Sea caused by wind conditions.
Materials and Methods. The study is based on the analysis of remote sensing data and observations of wind speed and direction over the Azov Sea. The primary method used is mathematical modelling, which includes solving the system of shallow water hydrodynamics equations. Wind condition data were collected from November 20 to 25, 2019, during which catastrophic sea level fluctuations were observed. The model considers the components of water flow velocity, water density, hydrodynamic pressure, gravitational acceleration, and turbulence exchange coefficients.
Results. The modelling showed that prolonged easterly winds with speeds up to 22 m/s led to significant surge and seiche fluctuations in sea level. The maximum amplitudes of fluctuations were recorded in the central part of the Taganrog Bay, where the wind direction and speed remained almost constant throughout the observation period. Data from various platforms located in different parts of the Azov Sea confirmed a significant decrease in water level in the northeast and an increase in the southwest.
Discussion and Conclusions. The study results confirm that using mathematical models in combination with remote sensing data allows more accurate predictions of extreme sea level fluctuations. This is important for developing measures to prevent and mitigate the consequences of surge and seiche events in coastal areas. In the future, it is necessary to improve models by including additional factors such as climate change and anthropogenic impact on the Azov Sea ecosystem.

Introduction. Currently, frequency modes of operation of electron accelerators based on capillary discharges are actively investigated. Electrons in these systems are accelerated by femtosecond laser pulses passing through the discharge plasma.
Materials and Methods. The paper presents results of three-dimensional magnetohydrodynamic modelling of the capillary discharge cycle, including stages of filling a short capillary with working gas (hydrogen), formation of the plasma channel, and restoration of the working medium before the start of the next discharge. Calculations were performed assuming the system is under external cooling, which maintains thermal balance at intermediate stages of the working cycle, and under constant conditions of gas supply and evacuation.
Results. The computational experiments demonstrate the capability of generating beams of relativistic electrons with a repetition frequency of approximately one kilohertz.
Discussion and Conclusions. The obtained results allow us to speak about the prospects of using LWFA with a short channel length and a high repetition rate of the capillary discharge.

Introduction. The study focuses on modelling the process of ultrasound medical examination in a heterogeneous environment with regions of significantly different sound speeds. Such scenarios typically arise when visualizing brain structures through the skull. The aim of this work is to compare possible approaches to determining the interface between acoustically contrasting media using convolutional neural networks.
Materials and Methods. Numerical modelling of the direct problem is performed, obtaining synthetic calculated ultrasonic images based on known geometry and rheology of the area as well as sensor parameters. The calculated images reproduce distortions and artifacts typical for setups involving the skull wall. Convolutional neural networks of 2D and 3D structures following the UNet architecture are used to solve the inverse problem of determining the interface between media based on a sensor signal. The networks are trained on computational datasets and then tested on individual samples not used in training.
Results. Numerical B-scans for characteristic setups were obtained. The possibility of localizing the aberrator boundary with good quality for both 2D and 3D convolutional networks was demonstrated. A higher quality result was obtained for the 3D network in the presence of significant noise and artifacts in the input data. It was established that the 3D architecture network can provide the shape of the interface between media in 0.1 seconds.
Discussion and Conclusions. The results can be used for the development of transcranial ultrasound technologies. Rapid localization of the skull boundary can be incorporated into imaging algorithms to compensate for distortions caused by differences in sound velocities in bone and soft tissues.

Introduction. Many mathematical problems are reduced to solving partial differential equations (PDEs) in domains of complex shapes. Existing analytical and numerical methods do not always provide efficient solutions for such problems. Recently, neural networks have been successfully applied to solve PDEs, typically addressing boundary value problems for domains with simple shapes. This paper attempts to construct a neural network capable of effectively solving boundary value problems for domains of complex shapes.
Materials and Methods. A method for constructing a neural network to solve the Dirichlet problem for regions of complex shape is proposed. Derivatives of singular solutions of the Laplace equation are accepted as activation functions. Singular points of these solutions are distributed along closed curves encompassing the boundary of the domain. The adjustment of the network weights is reduced to minimizing the root-mean-square error during training.
Results. The results of solving Dirichlet problems for various complex-shaped domains are presented. The results are provided in tables, comparing the exact solution and the solution obtained using the neural network. Figures show the domain shapes and the locations of points where the solutions were determined.
Discussion and Conclusion. The presented results indicate a good agreement between the obtained solution and the exact one. It is noted that this method can be easily applied to various boundary value problems. Methods for enhancing the efficiency of such neural networks are suggested.