Showing papers in "Russian Journal of Numerical Analysis and Mathematical Modelling in 2011"
TL;DR: In this article, the Frobenius norm is computed in O(d log q ε −1 ) time, where q ⩾ 2 is some fixed constant and d is the spatial dimension.
Abstract: Abstract In this paper QTT-approximations to elliptic solution operators with constant coefficients in d-dimensional cube are constructed. The ε-accurate representations of the Frobenius norm can be computed with the complexity O(d log q ε –1), where d ⩾ 2 is the spatial dimension, and q ⩾ 2 is some fixed constant.
47 citations
27 citations
TL;DR: In this paper, a model of a many-dimensional homogeneous isotropic random field with an exponential correlation function and a given arbitrary one-dimensional distribution suitable for numerical implementation is constructed.
Abstract: Abstract A model of a many-dimensional homogeneous isotropic random field with an exponential correlation function and a given arbitrary one-dimensional distribution suitable for numerical implementation is constructed. The model is based on partition of the phase space by a field of random hyperplanes determined by some parametric Poisson point field. The possibility of application of this model to the solution of stochastic problems of radiation balance based on the combination of the methods of ‘double randomization’ and ‘maximal section’ is discussed.
14 citations
TL;DR: In this article, the effect of an underwater slope unevenness on wave mode characteristics caused by the motion of a landslide over this slope is studied, and the results of the comparison of wave modes are discussed, the dependences of the characteristics of these modes on geometric and physical parameters of the studied phenomena, such as the landslide bedding depth, its length and thickness, the geometry of the slope and the friction force are analyzed.
Abstract: Abstract This paper is focused on the study of the effect of an underwater slope unevenness on the wave mode characteristics caused by the motion of a landslide over this slope. Using the simplest model representation of a landslide in the form of a rigid body, the authors consider two model reliefs, taking to some extent into account the peculiarities of the Mediterranean coast of Israel. The simulation of wave processes is performed within the framework of the equations of the shallow water theory. The results of the comparison of wave modes are discussed, the dependences of the characteristics of these modes on geometric and physical parameters of the studied phenomena, such as the landslide bedding depth, its length and thickness, the geometry of the slope, and the friction force are analyzed.
14 citations
TL;DR: In this article, a biased modification of the double local estimate of the Monte Carlo method is proposed, and a nonbiased modification with a finite variance is proposed to combine double local estimates.
Abstract: Abstract The problem of probing a medium with a ‘leader’ is considered in the paper. To do that, a biased modification of the ‘double local estimate’ of the Monte Carlo method is studied. In addition, its nonbiased modification with a finite variance (combined double local estimate) is proposed.
13 citations
TL;DR: In this paper, the initial boundary value problems are constructed for the shallow water equations in the form of series locally convergent in the neighbourhood of a movable water-land boundary for an arbitrary bottom relief.
Abstract: Abstract Solutions to initial boundary value problems are constructed for the shallow water equations in the form of series locally convergent in the neighbourhood of a movable water–land boundary for an arbitrary bottom relief. The motion law and the velocity of this boundary are determined for various wave–shore interaction modes. The obtained results of analytic study of the solutions are used for the development of approximations of boundary conditions on the movable shoreline. Test problems are numerically solved using an explicit predictor-corrector scheme of the second order of approximation on adaptive grids retracing the position of the water–land boundary. The results of these calculations are presented.
11 citations
TL;DR: In this article, a model of the joint circulation of the North Atlantic, the Arctic Ocean, and the Bering Sea is presented with the resolution of 0.25° in latitude and longitude.
Abstract: Abstract A model of the joint circulation of the North Atlantic, the Arctic Ocean, and the Bering Sea is presented with the resolution of 0.25° in latitude and longitude. The numerical technique of solving this problem and the organization of numerical experiments are described. Numerical calculations have been performed using this model for the period of 1958–2006. The results are compared with observation data and with the results of simulation by other models. Model estimates of the evolution of the Atlantic water incoming into the Arctic basin through the Fram Strait and the Barents Sea are presented. A positive trend of Atlantic water incoming into the Arctic basin through the Fram Strait is revealed. The evolution of the fresh water layer thickness in the Beaufort Gyre is considered. Three periods of increased thickness correlated with increased anticyclonic vorticity are pointed out: 1960s, 1980s, and the period from 1999 until now. The evolution of the anticyclonic vorticity is ahead of the changes in the fresh water layer thickness by 1.75 years. Long-term positive trends of fresh water layer thickness and the anticyclonic field vorticity in the Beaufort Gyre have been observed from the middle of 1970s. This period is characterized by a negative model trend of the ice area in the Arctic, which corresponds to observation data.
11 citations
TL;DR: In this article, a series of two-and three-dimensional stationary problems of a viscous incompressible fluid flow in a channel caused by a given pressure differential is studied, and it is shown that the solution to this problem is not unique.
Abstract: Abstract A series of two- and three-dimensional stationary problems of a viscous incompressible fluid flow in a channel caused by a given pressure differential is studied in the paper. It is shown that the solution to this problem is not unique. A numerical method allowing one to obtain the solution without specifying additional conditions for the velocity components on the boundary is considered.
9 citations
TL;DR: In this paper, the existence of periodic trajectories in phase portraits of odd-dimensional nonlinear dynamical systems modelling gene networks regulated by negative feedbacks has been proved, and sufficient conditions of the presence of stable cycles there have been given.
Abstract: Abstract We prove a theorem on the existence of periodic trajectories in phase portraits of odd-dimensional nonlinear dynamical systems modelling gene networks regulated by negative feedbacks. Also, we find certain sufficient conditions of the existence of stable cycles there. Some generalizations and applications of these results are given as well.
7 citations
TL;DR: In this paper, the problem of variational initialization of velocity, temperature, and salinity fields in the World Ocean is considered, where the ocean dynamics equations are written in the generalized σ-system of coordinates on a sphere with arbitrary positions of coordinate poles.
Abstract: Abstract The problem of four-dimensional variational initialization of velocity, temperature, and salinity fields in the World Ocean is considered. The ocean dynamics equations are written in the generalized σ-system of coordinates on a sphere with arbitrary positions of coordinate poles. The World Ocean model has the spatial resolution 2.5° × 2°× 33. The computational north pole is shifted onto the continent to the point (60° E, 60.5° N), the south pole coincides with the geographic one. The numerical experiments consist of two stages. At the first stage, we perform the calculations of the direct World Ocean circulation model for the period of 3000 years. On the ocean surface we specify the climatic atmospheric forcing constructed from the averaged CORE data from the period 1958–2004 with the discreteness of 6 hours. At the second stage, the problem is solved in the ‘variational initialization – forecast’ mode. The calculation interval equals 1 year, the solution obtained at the first stage is taken as the initial condition. After the initialization, the calculation of the direct model in the forecast mode is performed up to the end of the current month of the year. The mean monthly fields of temperature and salinity from the Argo buoys data for year 2008 are used as assimilated observation data. The results of the calculations show that observation data assimilation leads to a noticeable improvement of treatment characteristics. The model values approach the observations, and the solution adequately reflects the observed structure of natural fields.
6 citations
TL;DR: In this paper, an explicit method of the second order of accuracy (on smooth solutions) is proposed and implemented in Euler variables for the two-dimensional simulation of the indicated effect of non-axial breaking of relativistic and nonrelativistic cylindrical axially-symmetric nonlinear plasma oscillations.
Abstract: Abstract The effect of non-axial breaking of relativistic and nonrelativistic cylindrical axially-symmetric nonlinear plasma oscillations was earlier identified and then numerically and analytically studied by the authors within a one-dimensional hydrodynamic model. In this paper, an explicit method of the second order of accuracy (on smooth solutions) is proposed and implemented in Euler variables for the two-dimensional simulation of the indicated effect. The results of calculations admitting comparison with the one-dimensional model are presented.
TL;DR: In this paper, the problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function, and the optimal solution error is considered through the errors of input data (background and observation errors).
Abstract: Abstract The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The optimal solution error is considered through the errors of input data (background and observation errors). The optimal solution error covariance operator is approximated by the inverse Hessian of the auxiliary (linearized) data assimilation problem, which involves the tangent linear model constraints. We show that the derivative of the inverse Hessian with respect to the exact solution may be treated as the measure of nonlinearity for analysis error covariances in variational data assimilation problems.
TL;DR: In this paper, a mathematical model of the dynamics of oceans and seas is considered subject to tide-forming forces and the methods of description of tide-formation forces are discussed based on the variational data assimilation of satellite altimetry data, a solution algorithm for the inverse problem of reconstruction of selfattraction forces and a method for approximate solution of this problem are proposed.
Abstract: Abstract A mathematical model of the dynamics of oceans and seas is considered subject to tide-forming forces. The methods of description of tide-forming forces are discussed. Based on the variational data assimilation of satellite altimetry data, a solution algorithm for the inverse problem of reconstruction of ‘self-attraction’ forces and a method for approximate solution of this problem are proposed. The numerical study of the effect of tide-forming forces on the World Ocean dynamics is performed.
TL;DR: The σ-model of the general ocean circulation is described, its numerical implementation and the parallelization method for massively parallel computing systems are considered and the method called two-dimensional domain decomposition implemented by the MPI technology is used.
Abstract: Abstract The σ-model of the general ocean circulation is described in the paper, its numerical implementation and the parallelization method for massively parallel computing systems are considered. The method called two-dimensional domain decomposition implemented by the MPI technology is used. The efficiency of the code was tested on the ‘Lomonosov’ supercomputer at the Moscow University using a geographic ocean grid with the resolution of 0.5° × 0.25°.
TL;DR: In this article, the theory of scalar and vector probabilistic-algebraic algorithms of the Monte Carlo method used in the solution of systems of integral equations is detailed and refined.
Abstract: Abstract The theory of scalar and vector probabilistic–algebraic algorithms of the Monte Carlo method used in the solution of systems of integral equations is detailed and refined in the paper. A dual representation of the mean square of a vector estimate is constructed. Scalar algorithms are formulated, and a comparison of vector and scalar estimates of the solution is given for the first time. A criterion of finiteness of vector estimate variance is constructed on this basis.
TL;DR: Weighted Monte Carlo methods are proposed for the estimation of the functionals of the solutions to a Boltzmann type nonlinear equation in the kinetic model of a vehicle traffic flow with separated acceleration, which is related to a linear N-particle model of the vehicle system evolution.
Abstract: Abstract The problem of numerical estimation of the functionals of the solutions to a Boltzmann type nonlinear equation in the kinetic model of a vehicle traffic flow with separated acceleration is considered. The authors construct a second-kind integral equation for the original probabilistic model of a vehicle traffic flow, which is related to a linear N-particle model of the vehicle system evolution. Weighted Monte Carlo methods are proposed for the estimation of the functionals of the solution to the obtained equation. The practical suitability of this approach to the solution of traffic problems is demonstrated by numerical experiments. It should be noted that, in contrast to the previous papers, the authors do not use an artificial time step not included in the original traffic flow model.
TL;DR: In this paper, a local theory of symmetric one-step methods is presented, which is based on the Richardson technique only and does not require any asymptotic global error expansion.
Abstract: Abstract It is well known that the theory of asymptotic expansion of the global error of one-step methods is an important but complicated fact in the realm of numerical analysis of differential equations. It can hardly be confirmed for majority of numerical schemes and problems by practical computations. On the other hand, this theory is a fundamental tool to justify the extrapolation technique, which is one of the most efficient means to solve ordinary differential equations. Therefore, in the recent paper Kulikov [Numer. Algorithms 53: 321–342, 2010] presented his local theory of extrapolation methods, which is based on the Richardson technique only and does not require any asymptotic global error expansion. Here, we concentrate on quadratic extrapolation. We explain which property of symmetric one-step methods provides two-order growth of accuracy of the underlying method after each extrapolation step and arrive at the notion of proportional extrapolation. We also learn more about adjoint and symmetric one-step methods. In addition, we prove that the modified Aitken–Neville algorithm works for any symmetric one-step method of an arbitrary order 2s.
TL;DR: In this article, the effects caused by inexact knowledge of the Poisson's ratio in the linear elasticity problem were studied and the authors derived estimates for the incremental quantity that characterizes the variability of the internal energy and the logarithmic derivative of the energy with respect to the poisson ratio.
Abstract: We study the effects caused by inexact knowledge of the Poisson's ratio in the linear elasticity problem and derive estimates for the incremental quantity that characterizes the variability of the internal energy and the logarithmic derivative of the energy with respect to the Poisson's ratio. These quantities characterize errors caused by inexact (incomplete) knowledge of material constants, which is typical for computer modelling of physical and engineering problems. We prove that their be- haviour is drastically different for two different classes of boundary conditions, namely the boundary conditions of the first class generate solutions that are relatively stable with respect to small variations of the Poisson's ratio, while other conditions generate solutions, for which the quantities blow up if the ratio tends to one half.
TL;DR: Criteria are proposed for a priori determination whether sufficiently good approximation is guaranteed for the particular system in the case of approximation of the input and output signals by two Laguerre functions.
Abstract: Abstract Linear systems of ordinary differential and algebraic equations modelling passive integral circuits consisting of resistors and capacitors (RC-circuits) are considered. A method for the fast calculation of signal delay is such circuits is proposed. For this purpose, the input signal is approximated by several first Laguerre functions and the output of the system is obtained in the form of a sum of several Laguerre functions and one of the eigenvalues of the matrix pencil corresponding to the system (spectral correction of the solution). For the case of approximation of the input and output signals by two Laguerre functions, criteria are proposed for a priori determination whether sufficiently good approximation is guaranteed for the particular system. Results of numerical experiments with industrial design tests are presented and discussed.
TL;DR: In this article, a series of problems with free boundaries is considered using the ISPH method in the presence of break modes and the main attention is paid to the calculation of the pressure field and determination of hydrodynamic loads on the vertical walls of the basin.
Abstract: Abstract Numerical modelling of a series of problems with free boundaries is considered using the ISPH method in the presence of break modes. The main attention is paid to the calculation of the pressure field and determination of hydrodynamic loads on the vertical walls of the basin. The reliability of the results presented here is confirmed by the comparison with the results of other authors.
TL;DR: In this article, a model of stationary axisymmetric air motion in the lower layer of a typhoon is expounded and substantiated, which takes into account air viscosity and compressibility, as well as the Coriolis force and friction against the Earth surface.
Abstract: Abstract A model of stationary axisymmetric air motion in the lower layer of a typhoon is expounded and substantiated. The model takes into account air viscosity and compressibility, as well as the Coriolis force and friction against the Earth surface. The model is reduced to a system of three nonlinear differential equations for the horizontal velocity and density of air averaged in the vertical direction. Introducing the function β describing the vertical air flux at the top of the layer, the problem is decomposed into the two following ones solved successively: a boundary value problem for the tangential velocity component and a Cauchy problem for air density. After the solution of these problems, the remaining unknowns (the radial velocity component and pressure averaged in the vertical direction) are obtained explicitly. An efficient numerical implementation of the model is performed. The calculation results are in good accordance with observation data. The numerical experiments performed in this work reveal the existence of two maxima in the dependence of air velocity on the distance from the typhoon center in its lower layer.
TL;DR: In this article, the existence of the uniform attractor is established for the multilayer quasigeostrophic model of ocean dynamics with time-dependent forcing, and it is proved that when the mesh size decreases, the attractors of the schemes lie in an arbitrarily small neighbourhood of the actual attractor of the model.
Abstract: Abstract Semiexplicit and explicit finite-difference schemes for the multilayer quasigeostrophic model of ocean dynamics with time-dependent forcing are considered. The existence of the uniform attractor is established for each of the schemes. It is proved that when the mesh size decreases, the attractors of the schemes lie in an arbitrarily small neighbourhood of the actual attractor of the model.
TL;DR: In this paper, the authors developed numerical algorithms based on the natural element method for the solution of problems of incompressible fluid dynamics with free boundaries, where the authors focused on the development of numerical algorithms.
Abstract: Abstract This paper is focused on the development of numerical algorithms based on the natural element method for the solution of problems of incompressible fluid dynamics with free boundaries.
TL;DR: In this paper, a nonlinear difference scheme based on the classic approximation of the Dirichlet problem on a piecewise-uniform grid refined in a layer converges in the uniform norm with the convergence rate order not higher than one.
Abstract: Abstract Grid approximations of the Dirichlet problem are considered in a vertical strip for the semilinear elliptic convection–diffusion equation; for this problem, the nonlinear difference scheme based on the classic approximation of the problem on a piecewise-uniform grid refined in a layer converges ε-uniformly in the uniform norm with the convergence rate order not higher than one. Using the Richardson technique, we construct a nonlinear scheme (the Richardson scheme on nested piecewise-uniform grids) convergent ε-uniformly with the improved convergence rate , where N 1 +1 and N 2 +1 are the numbers of nodes on the axis x 1 and on the unit segment of the axis x 2, respectively. In order to solve this scheme, we construct iterative schemes of higher accuracy, which are the linearized scheme (the nonlinear term is calculated based on the desired function taken from the previous iteration) and the scheme of the Newton method, as well as truncated variants of iterative schemes convergent at the rate . The number T of iterations required in the truncated schemes for the solution of the problem does not depend on the parameter ε, here T = 𝒪(ln (min[N 1,N 2])) and T = 𝒪(ln ln (min[N 1, N 2])) in the case of the truncated linearized scheme and the truncated Newton scheme, respectively.
TL;DR: In this paper, a mathematical model of physical processes in a vortical flow created by magnetic hydrodynamics in a thin layer of a rotating viscous liquid is proposed.
Abstract: Abstract Mathematical models of physical processes in a vortical flow created by magnetic hydrodynamics in a thin layer of a rotating viscous liquid are proposed. An inverse problem on restoring the electric field vector is stated and studied, an algorithm of its numerical solution is formulated, and results of numerical experiments are presented.
TL;DR: In this article, Monte Carlo methods for the solution of the nonlinear kinetic Boltzmann equation are considered and a special two-parametric weighted method is constructed, similar to the method of majorant frequency and applicable to the case of an unbounded frequency of a pair collision.
Abstract: Abstract Monte Carlo methods for the solution of the nonlinear kinetic Boltzmann equation are considered. A special two-parametric weighted method is constructed. On the one hand, this method has improved performance, similar to the method of majorant frequency and, on the other hand, it is applicable to the case of an unbounded frequency of a pair collision.
TL;DR: In this article, the problem of interaction of a thermocouple inserted into a unitary solid propellant with a moving heat wave spreading inside the propellant from the combustion surface is considered.
Abstract: Abstract The problem of interaction of a thermocouple inserted into a unitary solid propellant with a moving heat wave spreading inside the propellant from the combustion surface is considered. The results of numerical modelling showed that a heat sink directed deep into the propellant occurs along the thermocouple wire because of large differences in the heat conductivity coefficients of the propellant and the thermocouple material, which essentially changes the temperature of the thermocouple junction and hence distorts the indications.
TL;DR: In this paper, an inverse problem of salinity fluxes and the corresponding problem of variational assimilation of ocean salinity data obtained from the international system of ARGO buoys are formulated and studied.
Abstract: An inverse problem of salinity fluxes and the corresponding problem of variational assimilation of ocean salinity data obtained from the international system of ARGO buoys are formulated and studied in the paper. A solution algorithm is developed for this problem. Results of numerical experiments are presented. One of the most important modern computational and information problems is the development of a set of computational algorithms solving many-dimensional nonstationary problems of variational data assimilation (observations obtained from satellites, measurements from sea vessels, etc.) and inverse problems in geophysical hydrodynamics. The solution of such problems is impeded by the complexity of mathematical models and domain boundaries (for example, real configuration of oceans and seas), a large number of discretization nodes, and a large amount of information needed to provide the considered mathematical models with data. Therefore, the statement and numerical solution of variational data assimilation problems for oceanographic observations is of great scientific interest. This paper is focused on the solution of an inverse problem of salinity fluxes in the World Ocean dynamics model and the corresponding problem of variational assimilation of data obtained from the ARGO buoys. Note that salinity is a basic hydrophysical parameter of the ocean, which affects sea water density and water-ice phase transitions, and also, to a certain extent, it can be an indicator related to such important characteristics as the amount of precipitation on a particular area and the evaporation rate on the ocean surface. Until recently, there were not enough data concerning the salinity of the World Ocean, and only the start of the international project of the ARGO profiling buoys system has made it possible to cover a sufficiently large water area of the World Ocean to obtain salinity data not only near the ocean surface, but within the depth of 1500 m. Now the amount of data allows one to perform full-scale numerical experiments for variational assimilation of observed salinity data of the World Ocean. ∗Institute of Numerical Mathematics of the Russian Academy of Sciences, Moscow 119993, Russia The work was supported by the Russian Foundation for Basic Research and by the Federal Target Program ‘Research and development on priority directions of scientific-technological complex of Russia for 2007–2013’.
TL;DR: In this paper, the problem of minimizing the P 1 interpolation error on a set of triangulations with a fixed number of triangles is reformulated as the minimization problem of generating a mesh which is quasi-uniform in a specially designed metric.
Abstract: Abstract For a given function, we consider the problem of minimizing the P 1 interpolation error on a set of triangulations with a fixed number of triangles. The minimization problem is reformulated as the problem of generating a mesh which is quasi-uniform in a specially designed metric. For functions with indefinite Hessian, we show the existence of a set of metrics with highly diverse properties. This set may include both anisotropic and isotropic metrics, which produce families of different meshes providing a comparable reduction of interpolation error. The developed theory is verified with numerical examples.