A review scientific work in the field of supercomputer technologies at the Institute of Mathematical Modelling and Keldysh Institute of Applied Mathematics is presented. Progress in supercomputer technologies, programming tools and technique (such as hyperbolization, load balancing, fault tolerance, adaptive mesh refinement, rational mesh decomposition) and several supercomputer applications are presented.

Introduction. To prevent the occurrence and mitigate the consequences of hazardous and catastrophic phenomena associated with sediment transport in natural systems, it is necessary to develop operational and scientifically justified forecasts, identify critical states at which the emergence of emergency situations is possible. For these purposes, it is necessary to create an accurate and efficient toolkit, including algorithms for numerical solution of a model problem that takes into account the specifics of natural systems. In this work, parallel algorithms for numerical solution of a spatially three-dimensional diffusion-convection problem of sediment are presented, which allow a significant reduction in computation time (by more than 4 times) compared to calculations conducted using a sequential algorithm.
Materials and Methods. For the parallel solution of the spatially three-dimensional diffusion-convection problem, an implicit splitting scheme is constructed, in which the original continuous problem is replaced by a chain of two-dimensional and one-dimensional problems. The splitting schemes proposed in the work are physically justified and take into account the specifics of coastal marine systems, for which the influence of micro-turbulent diffusion and advective transport of substances are comparable, and the Peclet number does not exceed unity when approximating real problems. For the parallel numerical implementation, a method of decomposing the grid domain into two families of vertical planes parallel to the coordinate planes Oxz and Oyz, combined with the Seidel method for solving two-dimensional grid problems in horizontal planes and the tridiagonal matrix algorithm when solving one-dimensional three-point problems in the vertical direction, is used. Within the framework of the parallel computing software implementation, a parallel algorithm is presented that implements the diffusion-convection problem on a computing system using MPI technology.
Results. A comparative analysis of parallel and sequential algorithms is obtained using a model problem.
Discussion and Conclusions. The developed software allows its practical use for solving specific hydrophysical problems, including as part of a software complex.

Introduction. Modern plasma magnetic confinement systems use tungsten as a material in contact with plasma. Under the influence of high-density plasma irradiation, tungsten undergoes cracking, intense erosion, and macro-particle emission. High-temperature ceramics are considered a promising material for protective coating of plasma components, as they are resistant to thermal loads. One possible solution could be a boron carbide coating, which has a high melting temperature.
Materials and Methods. The impact of an electron beam on samples of rolled tungsten and boron carbide and tungsten composite was studied in experiments on the BETA setup. The heat from the beam propagates into the samples, with the maximum temperature reached at the center and decreasing towards the edges. The modeling area represents a cross-section of the samples, optimal for a task with a cylindrical coordinate system. The numerical implementation is based on the correction scheme and the marching method.
Results. A new model of heating the boron carbide and tungsten composite sample under the influence of surface heating by an electron beam is presented. The model is based on solving the heat conduction equation in an axially symmetric setup with constant values of specific heat capacity, density, and thermal conductivity of metals.
Discussion and Conclusions. An analysis of the model of heating the composite material under the influence of surface heating by an electron beam at constant values of density, thermal conductivity, and specific heat capacity has been conducted. The modeling results are in demand for analyzing experimental results and planning experiments at the Beam of Electrons for Materials Test Applications (BETA) facility, created at the Budker Institute of Nuclear Physics SB RAS.

Introduction. We investigate a multidimensional (in terms of spatial variables) parabolic-type integro-differential equation with nonhomogeneous first-order boundary conditions. The locally one-dimensional finite difference scheme developed herein can be applied to solve various applied problems leading to multidimensional parabolic-type integro-differential equations. Examples include mathematical modelling of cloud processes, addressing the issue of active intervention in convective clouds to prevent hail and artificially enhance precipitation, as well as describing the droplet mass distribution function due to microphysical processes such as condensation, coagulation, fragmentation, and freezing of droplets in convective clouds.
Materials and Methods. In this study, an approximate solution to the initial-boundary value problem is constructed using the locally one-dimensional scheme of A.A. Samarsky with a specified order of approximation О(h² +τ). The primary research method employed is the method of energy inequalities.
Results. A priori estimates have been obtained in the discrete interpretation, from which uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow, with a convergence rate equal to the order of approximation of the difference scheme.
Discussion and Conclusions. The research findings can be utilized for further development of boundary value problem theory for parabolic equations with variable coefficients. Additionally, they may find applications in the fields of difference scheme theory, computational mathematics, and numerical modelling.

Introduction. The reduction in beach width due to erosion is a significant issue that can either be mitigated or exacerbated by coastal protection structures. Modelling breaking waves near the coast and around coastal structures can be used to determine their impact on the dynamics of the coastal zone. The objective of this study is to model and analyze the dynamics of turbulent structures around a single breakwater, obtained using two turbulence modelling schemes: RANS and LES.
Materials and Methods. Turbulence induced by breaking waves was investigated. The modelling was based on bathymetric measurements conducted along the Azov Sea coast and a three-dimensional wave hydrodynamics model supplemented with various turbulence calculation configurations.
Results. Modelling results of wave processes generating turbulent flows in the presence of coastal protection structures using different turbulence models were obtained. Results obtained based on Reynolds-averaged Navier-Stokes (RANS) equations are compared with the results of Large Eddy Simulation (LES) approach with Smagorinsky dynamic subgrid-scale model (DSM).
Discussion and Conclusions. The results showed that wave heights simulated by LES were higher than those simulated by RANS in the front and leeward regions of the coastal protection structure and were lower in its upper part. Thus, according to LES, a greater amount of wave energy was preserved after passing over the breakwater. Velocity vectors of the water medium showed the formation of a vortex when LES was used, whereas no evidence of such turbulent vortices was detected in the case of RANS, confirming the better performance of LES for turbulence modelling in the coastal zone. According to the presented results, LES is the preferred tool for generating turbulence under incoming wave conditions in engineering practices.