Pleiades Publishing Ltd

Numerical simulation and optimization of cryosurgeries are considered. The Stefan problem that describes thermal processes running in the course of surgery is solved numerically. The finite volume method and enthalpy approach are used. The problem of optimization of the probe positions is solved to improve the surgery efficiency. The optimization problem is solved using nonparametric multiple criteria optimization techniques.

A sequence converging to the solution of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation is constructed. This sequence is also asymptotic in the sense that the deviation (in the norm of the space of continuous functions) of its nth element from the solution of the problem is proportional to the (n + 1)th power of the perturbation parameter. A similar sequence is constructed for the case of an inhomogeneous first-order linear equation, on the example of which the application of such a sequence to the justification of the asymptotics obtained by the method of boundary functions is demonstrated.

The multioperator approach is used to obtain high-order accurate compact differences. These differences are developed to describe convective terms of differential equations, as well as mixed derivatives, source terms, and the coefficients of metric derivatives of coordinate transformations. The same principles are used to obtain high-order compact differences for representing diffusion terms. These differences underlie multioperator composite compact schemes, which are used to compute the flow past an airfoil by integrating the nonstationary Navier-Stokes equations supplemented with the equations of a turbulent viscosity model.

Explicit–implicit approximations are used to approximate nonstationary convection–diffusion equations in time. In unconditionally stable two-level schemes, diffusion is taken from the upper time level, while convection, from the lower layer. In the case of three time levels, the resulting explicit–implicit schemes are second-order accurate in time. Explicit alternating triangular (asymmetric) schemes are used for parabolic problems with a self-adjoint elliptic operator. These schemes are unconditionally stable, but conditionally convergent. Three-level modifications of alternating triangular schemes with better approximating properties were proposed earlier. In this work, two- and three-level alternating triangular schemes for solving boundary value problems for nonstationary convection–diffusion equations are constructed. Numerical results are presented for a two-dimensional test problem on triangular meshes, such as Delaunay triangulations and Voronoi diagrams.

A finite-difference method is used to prove the completeness of the eigenfunctions of the Sturm-Liouville operator in conservative form. The finite-difference schemes corresponding to the conservative Sturm-Liouville equation with various boundary conditions are shown to be self-adjoint. The accuracy and convergence of the method are analyzed, and the properties of eigenvalues and eigenvectors of the difference scheme approximating the differential equation and the boundary conditions are examined.

Some recent works related to the Rayleigh function are analyzed that reestablish results that have been known in the literature for over one hundred years.

A new numerical implementation of a fast-converging iterative method with splitting of boundary conditions is constructed for solving the Dirichlet initial-boundary value problem for the nonstationary Stokes system. The method was earlier proposed and substantiated at the differential level by B.V. Pal’tsev. The problem is considered in a strip and is assumed to be periodic along the strip. According to the numerical implementation proposed, a special vector parabolic problem for velocity approximations (which arises at iterations of the method) is discretized using an asymptotically stable two-stage difference scheme that is second-order accurate in time. The spatial discretization is based on bilinear finite elements on uniform rectangular grids. A numerical study shows that the convergence rate of the constructed iterative method is as high as that of the original method at the differential level (the error is reduced by approximately 7 times per iteration step). For velocities, the method is second-order accurate in the max norm. For pressures, the method is second-order accurate in space and first-order accurate in time.

Problems with inexactly specified parameters arise in many applications, and they are often formulated as dynamical systems with interval parameters. For such systems, it is required to know how to obtain an interval estimate of solution given the interval values of the parameters. The main idea underlying the adaptive interpolation algorithm is to construct an adaptive hierarchical grid based on the kd-tree over the set formed by the interval initial conditions and parameters of the problem. In this grid, each cell contains an interpolation grid. For each point in time, depending on the features of solution, the decomposition is adaptively rearranged. At each step, the algorithm produces a piecewise polynomial function that interpolates the dependence of the problem solution on the parameter values with prescribed accuracy. As the number of interval parameters grows, the number of nodes in the interpolation grid grows exponentially, which restricts the algorithm scope. To mitigate this drawback, it is proposed to use the TT-decomposition of the multidimensional arrays in which the values of nodes of the interpolation grids are stored instead of these arrays. The efficiency of the proposed modification of the algorithm is demonstrated on model problems. In particular, the combustion of hydrogen–oxygen mixture in the presence of indeterminacy in the constants of chemical reaction rates is simulated.

A new numerical algorithm based on multigrid methods is proposed for solving equations of the parabolic type. Theoretical error estimates are obtained for the algorithm as applied to a two-dimensional initial-boundary value model problem for the heat equation. The good accuracy of the algorithm is demonstrated using model problems including ones with discontinuous coefficients. As applied to initial-boundary value problems for diffusion equations, the algorithm yields considerable savings in computational work compared to implicit schemes on fine grids or explicit schemes with a small time step on fine grids. A parallelization scheme is given for the algorithm.