Introduction. Seismic exploration is a widely used technology for locating hydrocarbon deposits. An important stage of this process is the simulation of seismic wave propagation in a geological model of the medium with specified physical characteristics. Due to the high computational cost of this problem, the acoustic approximation is widely used in practice, allowing for the correct description of longitudinal wave propagation. The most common approach to seismic modeling is the use of finite-difference schemes on staggered Cartesian computational grids. Despite their simplicity of implementation and high computational efficiency, such methods exhibit insufficient accuracy when modelling complex geological structures, including curvilinear interfaces between geological layers. A promising direction is the development of new high-order computational methods on curvilinear computational grids. This paper presents a stable fifth-order grid-characteristic method successfully applied to solving the problem of acoustic wave propagation in the two-dimensional case.

Materials and Methods. The study employs a grid-characteristic method with a fifth-degree interpolation polynomial constructed on an extended spatial stencil. A class of curvilinear grids is identified that makes it possible to retain the accuracy achieved when solving a one-dimensional problem. Furthermore, the use of a multistage splitting method allows the preservation of the scheme’s order in both time and space for multidimensional formulations.

Results. The formulas of the computational algorithm are presented, the achievement of the declared convergence order is empirically confirmed, and wavefield patterns of the dynamic process are calculated.

Discussion. The results demonstrate lower numerical dissipation of the proposed computational algorithm. The trade-off for this improvement is a significant increase in computation time.

Conclusion. The developed computational algorithm ensures high accuracy in calculating seismic fronts, which is critically important for seismic exploration tasks in layered geological massifs.

Introduction. A finite-difference scheme approximating a boundary value problem for a parabolic-type equation in a three-dimensional setting with boundary conditions of the first-third types is considered. This paper is a continuation of the authors’ previous works devoted to the numerical solution of one of the pressing problems of hydrophysics in shallow marine zones – the problem of transport, deposition, and transformation of suspended matter. The approximation of this lass of problems inside the domain leads to schemes converging at a rate of O(τ + h²), where h² = h²_x + h²_y + h²_z, h_x, h_y, h_z and τ are the steps of the difference grid along the spatial coordinates x, y, z and time, respectively. However, the case of boundary conditions requires careful treatment, since an inaccurate approximation may reduce the overall order of accuracy of the finite-difference scheme. The methods proposed by the authors for approximating boundary conditions ensure the convergence of the finite-difference scheme at the rate of O(τ + h²).

Materials and Methods. The authors focused on approximating third-type boundary conditions (with second-type conditions considered as a particular case). The approach is based on the central difference approximation of boundary conditions on an extended grid and the elimination of suspended matter concentration values in ghost nodes (cells).

Results. Approximations of the second- and third-type boundary conditions were constructed for a boundary value problem describing suspended matter transport. These approximations guarantee convergence of the finite-difference scheme at the rate of O(τ + h²).

Discussion. The study may be useful in convection–diffusion problems where achieving numerical solutions with acceptable accuracy is required.

Conclusion. Future research may focus on the analysis of the constructed finite-difference schemes under physically motivated constraints on the time step τ and the grid Peclet number.

Introduction. A two-dimensional hydrodynamic problem in the «stream function–vorticity» variables is numerically solved in an open rectangular cavity simulating blood flow in a blood vessel aneurysm. Two solution algorithms are proposed for Reynolds numbers Re < 1 and for Re ≥ 1.

Materials and Methods. To accelerate the numerical solution with an explicit finite-difference scheme for the vorticity dynamics equation, the initial condition damping method, the n-fold splitting method of the explicit finite-differences cheme (n = 100, 200), and the symmetry plane of the rectangular cavity–aneurysm were employed. In the splitting method, the maximum time step proportional to the square of the spatial step was used without violating the spectral stability of the explicit scheme in the vorticity equation. On half of the rectangular aneurysm, symmetric solutions were considered with a uniform 100 × 50 grid and equal steps h₁ = h₂ = 0.01. The inverse matrix for solving the Poisson equation in the «stream function–vorticity» variables with a finite number of elementary operations was computed using the MSIMSL library.

Results. The numerical solution showed that the number and location of circulation regions in the aneurysm at small Reynolds numbers depend on the ratio of the vessel diameter to the aneurysm diameter. At small values of this parameter, the aneurysm contains a single large vortex that narrows the vessel lumen in the case of thrombus formation inside the aneurysm. The narrowing of the blood flow tube inside the aneurysm reaches 34%. It was found that the formation of the hydrodynamic structure in the aneurysm occurs in a time negligible (0.002%) compared to the period between pulsation waves (1 s). For the first time, a boundary condition with fourth-order accuracy was proposed to relate velocity, vorticity, and stream function.

Discussion. The approximation of the equations in systems (4) and (22) has sixth-order accuracy at interior nodes and fourth-order accuracy at boundary nodes. The problem was also solved for blood motion in arteries at high Reynolds numbers (Re = 1500). The solution shows that in the aneurysm symmetry plane a chain of connected vortices is formed with alternating signs of vorticity, carried by the blood flow along the vessel.

Conclusion. The initial–boundary value problems (4), (22) formulated in this work make it possible to qualitatively model blood flow in aneurysms of capillaries, arterioles, and arteries at low and high velocities, as well as blood motion in elements of medical equipment.

Introduction. Shallow coastal systems are highly dynamic and require accurate numerical models for predicting tides, storm surges, and coastal hazards. Traditional uniform-grid approaches often incur high computational costs, limiting their applicability for operational forecasting. Adaptive grid techniques provide a promising alternative by concentrating resolution in dynamically important regions while reducing the total computational burden.

Materials and Methods. We developed an adaptive-grid framework based on the depth-averaged shallow-water equations. The model employs a second-order finite-volume scheme with TVD limiting on a quadtree mesh. Mesh adaptation is driven by gradient indicators of free-surface elevation and velocity, ensuring high resolution in areas with steep gradients, tidal fronts, and complex bathymetry. Three numerical experiments were performed: a harmonic tide, a wind-driven storm surge, and combined tidal-wind forcing.

Results. The proposed method demonstrated robust wetting-drying capabilities, a mass conservation error below 0.06%, and skill metrics of RMSE ≤ 0.07 m and NSE ≥ 0.90. Compared to a uniform grid of the same finest resolution, Adaptive Mesh Refinement (AMR) reduced the mean cell count by ~32% and wall time by ~1.5×, with less than 3.5% change in the L₂ error norm.

Discussion. The results confirm that adaptive meshing preserves physical accuracy while substantially reducing computational cost. This makes the method a suitable tool for high-resolution coastal hazard assessment and operational forecasting.

Conclusion. Further work will focus on extending the approach to three-dimensional flows and incorporating data assimilation for real-time applications.

Introduction. In recent years, the field of mathematics specializing in the application of artificial neural networks has been rapidly developing. In this work, a new method for constructing a neural network for solving wave differential equations is proposed. This method is particularly effective in solving boundary value problems for domains of complex geometric shapes.

Materials and Methods. A method is proposed for constructing a neural network designed to solve the wave equation in a planar domain G bounded by an arbitrary closed curve. It is assumed that the boundary conditions are periodic functions of time t, and the steady-state regime is considered. When constructing the neural network, the activation functions are taken as derivatives of singular solutions of the Helmholtz equation. The singular points of these solutions are uniformly distributed along closed curves surrounding the domain boundary. The training set consists of a set of particular solutions of the Helmholtz equation.

Results. Results were obtained for the solution of the first boundary value problem in various domains of complex geometric shape and under different boundary conditions. The results are presented in tables containing both the exact solutions of the problem and the solutions obtained using the neural network. A graphical comparison is also provided between the exact solution and the solution obtained with the constructed neural network.

Discussion. The presented computational results demonstrate the efficiency of the proposed method for constructing neural networks that solve boundary value problems of partial differential equations in domains of complex geometry.

Conclusion. The further development of the proposed method may be applied to solving boundary value problems for the wave equation in exterior domains. Of particular interest is the application of this method to diffraction problems.