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We consider a model of stacked long Josephson junctions (LJJ), which consists of alternating superconducting and dielectric layers. The model takes into account the inductive and capacitive coupling between the neighbor junctions. The model is described by a system of nonlinear partial differential equations with respect to the phase differences and the voltage of LJJ, with appropriate initial and boundary conditions. The numerical solution of this system of equations is based on the use of standard three-point finite-difference formulae for discrete approximations in the space coordinate, and the applying the four-step Runge-Kutta method for solving the Cauchy problem obtained. Designed parallel algorithm is implemented by means of the MPI technology (Message Passing Interface). In the paper, the mathematical formulation of the problem is given, numerical scheme and a method of calculation of the current-voltage characteristics of the LJJ system are described. Two variants of parallel implementation are presented. The influence of inductive and capacitive coupling between junctions on the structure of the current-voltage characteristics is demonstrated. The results of methodical calculations with various parameters of length and number of Josephson junctions in the LJJ stack depending on the number of parallel computing nodes, are presented. The calculations have been performed on multiprocessor clusters HybriLIT and CICC of Multi-Functional Information and Computing Complex (Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna). The numerical results are discussed from the viewpoint of the effectiveness of presented approaches of the LJJ system numerical simulation in parallel. It has been shown that one of parallel algorithms provides the 9 times speedup of calculations.

New family of linearly implicit schemes are presented. This family allows to obtain methods which are equivalent to stiffly accurate implicit Runge–Kutta schemes (such as RadauIIA and LobattoIIIC) on nonautonomous linear problems. Notion of LN-equivalence of schemes is introduced. Order conditions and stability conditions of such methods are obtained with the use of media for computer symbolic calculations. Some examples of new schemes have been constructed. Numerical studying of new method have been done with the use of classical tests for stiff problems.

The results of the numerical investigations of pulsating detonation wave propagation using the ENO-schemes with the approximation orders from the first to the fourth inclusively are presented. The results obtained with the use of the schemes of different approximation orders demonstrate that the pattern of detonation wave propagation in acetylene-air mixture corresponds to the analytical estimates both qualitatively and quantitatively. For the hydrogen-air mixture none of the schemes concerned provides the stable detonation wave propagation. The transition from the regular mode to the marginal one with the subsequent detonation breakup is observed.

Currently, different nonlinear numerical schemes of the spatial approximation are used in numerical simulation of boundary value problems for hyperbolic systems of partial differential equations (e. g. gas dynamics equations, MHD, deformable rigid body, etc.). This is due to the need to improve the order of accuracy and perform simulation of discontinuous solutions that are often occurring in such systems. The need for non-linear schemes is followed from the barrier theorem of S. K. Godunov that states the impossibility of constructing a linear scheme for monotone approximation of such equations with approximation order two or greater. One of the most accurate non-linear type schemes are ENO (essentially non oscillating) and their modifications, including WENO (weighted, essentially non oscillating) scemes. The last received the most widespread, since the same stencil width has a higher order of approximation than the ENO scheme. The benefit of ENO and WENO schemes is the ability to maintain a high-order approximation to the areas of non-monotonic solutions. The main difficulty of the analysis of such schemes comes from the fact that they themselves are nonlinear and are used to approximate the nonlinear equations. In particular, the linear stability condition was obtained earlier only for WENO5 scheme (fifth-order approximation on smooth solutions) and it is a numerical one. In this paper we consider the problem of construction and stability for WENO5, WENO7, WENO9, WENO11, and WENO13 finite volume schemes for the Hopf equation. In the first part of this article we discuss WENO methods in general, and give the explicit expressions for the coefficients of the polynomial weights and linear combinations required to build these schemes. We prove a series of assertions that can make conclusions about the order of approximation depending on the type of local solutions. Stability analysis is carried out on the basis of the principle of frozen coefficients. The cases of a smooth and discontinuous behavior of solutions in the field of linearization with frozen coefficients on the faces of the final volume and spectra of the schemes are analyzed for these cases. We prove the linear stability conditions for a variety of Runge-Kutta methods applied to WENO schemes. As a result, our research provides guidance on choosing the best possible stability parameter, which has the smallest effect on the nonlinear properties of the schemes. The convergence of the schemes is followed from the analysis.

We present the iterative algorithm that solves numerically both Urysohn type Fredholm and Volterra nonlinear one-dimensional nonsingular integral equations of the second kind to a specified, modest user-defined accuracy. The algorithm is based on descending recursive sequence of quadratures. Convergence of numerical scheme is guaranteed by fixed-point theorems. Picard’s method of integrating successive approximations is of great importance for the existence theory of integral equations but surprisingly very little appears on numerical algorithms for its direct implementation in the literature. We show that successive approximations method can be readily employed in numerical solution of integral equations. By that the quadrature algorithm is thoroughly designed. It is based on the explicit form of fifth-order embedded Runge–Kutta rule with adaptive step-size self-control. Since local error estimates may be cheaply obtained, continuous monitoring of the quadrature makes it possible to create very accurate automatic numerical schemes and to reduce considerably the main drawback of Picard iterations namely the extremely large amount of computations with increasing recursion depth. Our algorithm is organized so that as compared to most approaches the nonlinearity of integral equations does not induce any additional computational difficulties, it is very simple to apply and to make a program realization. Our algorithm exhibits some features of universality. First, it should be stressed that the method is as easy to apply to nonlinear as to linear equations of both Fredholm and Volterra kind. Second, the algorithm is equipped by stopping rules by which the calculations may to considerable extent be controlled automatically. A compact C++-code of described algorithm is presented. Our program realization is self-consistent: it demands no preliminary calculations, no external libraries and no additional memory is needed. Numerical examples are provided to show applicability, efficiency, robustness and accuracy of our approach.

This work is dedicated to application of bicompact schemes to numerical solution of evolutionary hyperbolic equations. The main advantage of this class of schemes lies in combination of two beneficial properties: the first one is spatial approximation of high even order on a stencil that always occupies only one mesh cell; the second one is spectral resolution which is better in comparison to classic compact finite-difference schemes of the same order of spatial approximation. One feature of bicompact schemes is considered: their spatial approximation is rigidly tied to Cartesian meshes (with parallelepiped-shaped cells in three-dimensional case). This feature makes rather challenging any application of bicompact schemes to problems with complex computational domains as treated in the framework of unstructured meshes. This problem is proposed to be solved using well-known methods for treating complex-shaped boundaries and their corresponding boundary conditions on Cartesian meshes. The generalization of bicompact schemes on problems in geometrically complex domains is made in case of gas dynamics problems and Euler equations. The free boundary method is chosen as a particular tool to introduce the influence of arbitrary-shaped solid boundaries on gas flows on Cartesian meshes. A brief description of this method is given, its governing equations are written down. Bicompact schemes of fourth order of approximation in space with locally one-dimensional splitting are constructed for equations of the free boundary method. Its compensation flux is discretized with second order of accuracy. Time stepping in the obtained schemes is done with the implicit Euler method and the third order accurate $L$-stable stiffly accurate three-stage singly diagonally implicit Runge–Kutta method. The designed bicompact schemes are tested on three two-dimensional problems: stationary supersonic flows with Mach number three past one circular cylinder and past three circular cylinders; the non-stationary interaction of planar shock wave with a circular cylinder in a channel with planar parallel walls. The obtained results are in a good agreement with other works: influence of solid bodies on gas flows is physically correct, pressure in control points on solid surfaces is calculated with the accuracy appropriate to the chosen mesh resolution and level of numerical dissipation.

The work is devoted to numerical modeling of two-phase flows, namely, the calculation of supersonic flow around a blunt body by a viscous gas flow with an admixture of large high inertia particles. The system of unsteady Navier – Stokes equations is numerically solved by the meshless method. It uses the cloud of points in space to represent the fields of gas parameters. The spatial derivatives of gas parameters and functions are approximated by the least square method to calculate convective and viscous fluxes in the Navier – Stokes system of equations. The convective fluxes are calculated by the HLLC method. The third-order MUSCL reconstruction scheme is used to achieve high order accuracy. The viscous fluxes are calculated by the second order approximation scheme. The streamlined body surface is represented by a model of an isothermal wall. It implements the conditions for the zero velocity and zero pressure gradient, which is also modeled using the least squares method.

Every moving body is surrounded by its own cloud of points belongs to body’s domain and moving along with it in space. The explicit three-sage Runge–Kutta method is used to solve numerically the system of gas dynamics equations in the main coordinate system and local coordinate systems of each particle.

Two methods for the moving objects modeling with reverse impact on the gas flow have been implemented. The first one uses stationary point clouds with fixed neighbors within the same domain. When regions overlap, some nodes of one domain, for example, the boundary nodes of the particle domain, are excluded from the calculation and filled with the values of gas parameters from the nearest nodes of another domain using the least squares approximation of gradients. The internal nodes of the particle domain are used to reconstruct the gas parameters in the overlapped nodes of the main domain. The second method also uses the exclusion of nodes in overlapping areas, but in this case the nodes of another domain take the place of the excluded neighbors to build a single connected cloud of nodes. At the same time, some of the nodes are moving, and some are stationary. Nodes membership to different domains and their relative speed are taken into account when calculating fluxes.

The results of modeling the motion of a particle in a stationary gas and the flow around a stationary particle by an incoming flow at the same relative velocity show good agreement for both presented methods.

The numerical solving of the system of high-temperature radiative gas dynamics (HTRGD) equations is a computationally laborious task, since the interaction of radiation with matter is nonlinear and non-local. The radiation absorption coefficients depend on temperature, and the temperature field is determined by both gas-dynamic processes and radiation transport. The method of splitting into physical processes is usually used to solve the HTRGD system, one of the blocks consists of a joint solving of the radiative transport equation and the energy balance equation of matter under known pressure and temperature fields. Usually difference schemes with orders of convergence no higher than the second are used to solve this block. Due to computer memory limitations it is necessary to use not too detailed grids to solve complex technical problems. This increases the requirements for the order of approximation of difference schemes. In this work, bicompact schemes of a high order of approximation for the algorithm for the joint solution of the radiative transport equation and the energy balance equation are implemented for the first time. The proposed method can be applied to solve a wide range of practical problems, as it has high accuracy and it is suitable for solving problems with coefficient discontinuities. The non-linearity of the problem and the use of an implicit scheme lead to an iterative process that may slowly converge. In this paper, we use a multiplicative HOLO algorithm named the quasi-diffusion method by V.Ya.Goldin. The key idea of HOLO algorithms is the joint solving of high order (HO) and low order (LO) equations. The high-order equation (HO) is the radiative transport equation solved in the energy multigroup approximation, the system of quasi-diffusion equations in the multigroup approximation (LO₁) is obtained by averaging HO equations over the angular variable. The next step is averaging over energy, resulting in an effective one-group system of quasi-diffusion equations (LO₂), which is solved jointly with the energy equation. The solutions obtained at each stage of the HOLO algorithm are closely related that ultimately leads to an acceleration of the convergence of the iterative process. Difference schemes constructed by the method of lines within one cell are proposed for each of the stages of the HOLO algorithm. The schemes have the fourth order of approximation in space and the third order of approximation in time. Schemes for the transport equation were developed by B.V. Rogov and his colleagues, the schemes for the LO₁ and LO₂ equations were developed by the authors. An analytical test is constructed to demonstrate the declared orders of convergence. Various options for setting boundary conditions are considered and their influence on the order of convergence in time and space is studied.

In this article, a meshfree-spectral method for numerical investigation of dynamics of planar geophysical flows is proposed. We investigate inviscid incompressible fluid flows with the presence of planetary rotation. Mathematically this problem is described by the non-steady system of two partial differential equations in terms of stream and vorticity functions with different boundary conditions (closed flow region and periodic conditions). The proposed method is based on several assumptions. First of all, the vorticity field is given by its values on the set of particles. The function of vorticity distribution is approximated by piecewise cubic polynomials. Coefficients of polynomials are found by least squares method. The stream function is calculated by using the spectral global Bubnov –Galerkin method at each time step.

The dynamics of fluid particles is calculated by pseudo-symplectic Runge –Kutta method. A detailed version of the method for periodic boundary conditions is described in this article for the first time. The adequacy of numerical scheme was examined on test examples. The dynamics of the configuration of four identical circular vortex patches with constant vorticity located at the vertices of a square with a center at the pole is investigated by numerical experiments. The effect of planetary rotation and the radius of patches on the dynamics and formation of vortex structures is studied. It is shown that, depending on the direction of rotation, the Coriolis force can enhance or slow down the processes of interaction and mixing of the distributed vortices. At large radii the vortex structure does not stabilize.