Showing papers in "Computational Mathematics and Mathematical Physics in 2015"

Internal layers in the one-dimensional reaction–diffusion equation with a discontinuous reactive term

Abstract: A singularly perturbed boundary value problem for a second-order ordinary differential equation known in applications as a stationary reaction–diffusion equation is studied. A new class of problems is considered, namely, problems with nonlinearity having discontinuities localized in some domains, which leads to the formation of sharp transition layers in these domains. The existence of solutions with an internal transition layer is proved, and their asymptotic expansion is constructed.

On the elasticity of blood vessels in one-dimensional problems of hemodynamics

Abstract: One-dimensional models of hemodynamics proved to be effective in the analysis of blood flow in humans in the normal and pathological states. A key factor contributing to the successful simulation using one-dimensional models is the inclusion of elastic properties of blood vessel walls. This paper is devoted to the comparative analysis of various mathematical descriptions of elastic properties of vessel walls in modern one-dimensional models of hemodynamics.

A mathematical model of pollutant propagation in near-ground atmospheric layer of a coastal region and its software implementation

Abstract: Mathematical modeling of pollutant propagation in the air of coastal regions is considered. A new mathematical model of aerodynamic processes is proposed that takes into account a multitude of factors acting in coastal regions, such as high air humidity, variability of the air pressure and temperature, etc. Algorithms for the investigation of this model are developed and their software implementation is described.

A randomized algorithm for two-cluster partition of a set of vectors

Abstract: A randomized algorithm is substantiated for the strongly NP-hard problem of partitioning a finite set of vectors of Euclidean space into two clusters of given sizes according to the minimum-of-the sum-of-squared-distances criterion. It is assumed that the centroid of one of the clusters is to be optimized and is determined as the mean value over all vectors in this cluster. The centroid of the other cluster is fixed at the origin. For an established parameter value, the algorithm finds an approximate solution of the problem in time that is linear in the space dimension and the input size of the problem for given values of the relative error and failure probability. The conditions are established under which the algorithm is asymptotically exact and runs in time that is linear in the space dimension and quadratic in the input size of the problem.

Linear instability of solutions in a mathematical model describing polymer flows in an infinite channel

Abstract: A new rheological model describing flows of incompressible viscoelastic polymer fluid is studied. Lyapunov’s linear instability of the Poiseuille flow counterpart for the Navier-Stokes equations in a plane infinite channel is proven.

Computational identification of the right-hand side of a parabolic equation

Abstract: Among inverse problems for partial differential equations, a task of interest is to study coefficient inverse problems related to identifying the right-hand side of an equation with the use of additional information. In the case of nonstationary problems, finding the dependence of the right-hand side on time and the dependence of the right-hand side on spatial variables can be treated as independent tasks. These inverse problems are linear, which considerably simplifies their study. The time dependence of the right-hand side of a multidimensional parabolic equation is determined using an additional solution value at a point of the computational domain. The inverse problem for a model equation in a rectangle is solved numerically using standard spatial difference approximations. The numerical algorithm relies on a special decomposition of the solution whereby the transition to a new time level is implemented by solving two standard grid elliptic problems. Numerical results are presented.

Particle method for electrons in a scattering medium

Abstract: The Cauchy problem for the kinetic and electrodynamic equations describing the propagation of an electron flow in a scattering medium and generation of self-consistent electromagnetic field is considered. The electron distribution function is defined in the space of finitely supported generalized functions. Algorithms for the simulation of scattering in the approximation of single and multiple collisions in a time step are presented. Specificities of application of this algorithm in a dense scattering medium and ionized region of large volume are considered.

Complex conservative difference schemes for computing supersonic flows past simple aerodynamic forms

Abstract: Complex conservative modifications of two-dimensional difference schemes on a minimum stencil are presented for the Euler equations. The schemes are conservative with respect to the basic divergent variables and the divergent variables for spatial derivatives. Approximations of boundary conditions for computing flows around variously shaped bodies (plates, cylinders, wedges, cones, bodies with cavities, and compound bodies) are constructed without violating the conservation properties in the computational domain. Test problems for computing flows with shock waves and contact discontinuities and supersonic flows with external energy sources are described.

Estimates of the Hyperbolization Effect on the Heat Equation

Abstract: The difference between the solutions of the heat equation and its hyperbolized version is estimated. The estimates are obtained in the L2 norm for the anisotropic heat equation and in the C norm for the one-dimensional case with constant coefficients.

Fast method for verifying Chernikov rules in Fourier-Motzkin elimination

Abstract: The problem of eliminating unknowns from a system of linear inequalities is considered. A new fast technique for verifying Chernikov rules in Fourier-Motzkin elimination is proposed, which is an adaptation of the “graph” test for adjacency in the double description method. Numerical results are presented that confirm the effectiveness of this technique.

On an algorithm for solving parabolic and elliptic equations

Abstract: The present-day rapid growth of computer power, in particular, parallel computing systems of ultrahigh performance requires a new approach to the creation of models and solution algorithms for major problems. An algorithm for solving parabolic and elliptic equations is proposed. The capabilities of the method are demonstrated by solving astrophysical problems on high-performance computer systems with massive parallelism.

Stability of nonstationary solutions of the generalized KdV-Burgers equation

Abstract: The stability of nonstationary solutions to the Cauchy problem for a model equation with a complex nonlinearity, dispersion, and dissipation is analyzed. The equation describes the propagation of nonlinear longitudinal waves in rods. Previously, complex behavior of traveling waves was found, which can be treated as discontinuity structures in solutions of the same equation without dissipation and dispersion. As a result, the solutions of standard self-similar problems constructed as a sequence of Riemann waves and shocks with a stationary structure become multivalued. The multivaluedness of the solutions is attributed to special discontinuities caused by the large effect of dispersion in conjunction with viscosity. The stability of special discontinuities in the case of varying dispersion and dissipation parameters is analyzed numerically. The computations performed concern the stability analysis of a special discontinuity propagating through a layer with varying dispersion and dissipation parameters.

Eddy-resolving numerical scheme for simulation of turbulent incompressible flows

Abstract: The properties of existing numerical schemes for discretizing convective fluxes are examined. An alternative scheme is proposed that is formulated on the basis of normalized variable diagrams. The dissipation properties of this scheme are analyzed by simulating turbulent incompressible flows with the use of eddy-resolving models. It is shown that the scheme does not lead to spurious oscillations and it is superior in dissipation properties to similar central-difference schemes. For the scheme proposed, an LES model is calibrated for both free and near-wall flows and the numerical solution of a plane channel flow problem is shown to converge by applying direct numerical simulation of turbulence.

Sparse representation of system of Fredholm integro-differential equations by using alpert multiwavelets

Abstract: A numerical technique is presented for the solution of system of Fredholm integro-differential equations. The method consists of expanding the required approximate solution as the elements of Alpert multiwavelet functions (see Alpert B. et al. J. Comput. Phys. 2002, vol. 182, pp. 149–190). Using the operational matrix of integration and wavelet transform matrix, we reduce the problem to a set of algebraic equations. This system is large. We use thresholding to obtain a new sparse system; consequently, GMRES method is used to solve this new system. Numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces accurate results.

Numerical computation of wave propagation in fractured media by applying the grid-characteristic method on hexahedral meshes

Abstract: Wave propagation in fractured rock in the course of seismic exploration is studied. The grid-characteristic method on hexahedral meshes is extended to the case of an elastic medium with empty and fluid-saturated cracks. The crack effect on wave propagation in the medium is taken into account by introducing cracks at the stage of grid generation with boundary conditions and conditions on the crack edges specified in explicit form. This method is used to obtain wave patterns near an extended inclined crack. The problem of numerically computing the seismic effect produced by a cluster of vertical and subvertical cracks is given in a complete three-dimensional formulation. The structure of the resulting pattern and the influence exerted by the crack-filling substance on the signal recorded on the surface are examined.

Subdomain finite element method with quartic B-splines for the modified equal width wave equation

Abstract: In this paper, a numerical solution of the modified equal width wave (MEW) equation, has been obtained by a numerical technique based on Subdomain finite element method with quartic B-splines. Test problems including the motion of a single solitary wave and interaction of two solitary waves are studied to validate the suggested method. Accuracy and efficiency of the proposed method are discussed by computing the numerical conserved laws and error norms L 2 and L ∞. A linear stability analysis based on a Fourier method shows that the numerical scheme is unconditionally stable.

Generalized functions and additional boundary conditions in heat conduction problems for multilayered bodies

Abstract: The basic principles of a method for finding approximate analytical solutions of nonstationary heat conduction problems for multilayered structures are described. The method relies on determining a temperature perturbation front and introducing additional boundary conditions. An asymmetric unit step function is used to represent the original multilayered system as a single-layer one with piecewise homogeneous medium properties. Due to the splitting of the heat conduction process into two stages, the original partial differential equation is reduced at each stage to solving an ordinary differential equation. As a result, fairly simple (in form) analytical solutions are obtained with accuracy depending on the number of specified additional boundary conditions (on the number of approximations). It is shown that, as the number of approximations increases, same-type ordinary differential equations are obtained for the unknown time functions at the first and second stages of the process. As a result, analytical solutions can be found with a nearly prescribed degree of accuracy, including small and supersmall times.

On the Efficiency of a Randomized Mirror Descent Algorithm in Online Optimization Problems

Abstract: A randomized online version of the mirror descent method is proposed. It differs from the existing versions by the randomization method. Randomization is performed at the stage of the projection of a subgradient of the function being optimized onto the unit simplex rather than at the stage of the computation of a subgradient, which is common practice. As a result, a componentwise subgradient descent with a randomly chosen component is obtained, which admits an online interpretation. This observation, for example, has made it possible to uniformly interpret results on weighting expert decisions and propose the most efficient method for searching for an equilibrium in a zero-sum two-person matrix game with sparse matrix.

Embedded symmetric nested implicit Runge–Kutta methods of Gauss and Lobatto types for solving stiff ordinary differential equations and Hamiltonian systems

Abstract: A technique for constructing nested implicit Runge–Kutta methods in the class of mono-implicit formulas of this type is studied. These formulas are highly efficient in practice, since the dimension of the original system of differential equations is preserved, which is not possible in the case of implicit multistage Runge–Kutta formulas of the general from. On the other hand, nested implicit Runge-Kutta methods inherit all major properties of general formulas of this form, such as A-stability, symmetry, and symplecticity in a certain sense. Moreover, they can have sufficiently high stage and classical orders and, without requiring high extra costs, can ensure dense output of integration results of the same accuracy as the order of the underlying method. Thus, nested methods are efficient when applied to the numerical integration of differential equations of various sorts, including stiff and nonstiff problems, Hamiltonian systems, and invertible equations. In this paper, previously proposed nested methods based on the Gauss quadrature formulas are generalized to Lobatto-type methods. Additionally, a unified technique for constructing all such methods is proposed. Its performance is demonstrated as applied to embedded examples of nested implicit formulas of various orders. All the methods constructed are supplied with tools for local error estimation and automatic variable-stepsize mesh generation based on an optimal stepsize selection. These numerical methods are verified by solving test problems with known solutions. Additionally, a comparative analysis of these methods with Matlab built-in solvers is presented.

A new version of the discrete ordinate method for the calculation of the intrinsic radiation in horizontally homogeneous atmospheres

Abstract: A new version of the discrete ordinate method for the calculation of the transfer of monochromatic radiation in a scattering, absorbing, and emitting plane-parallel atmosphere of Earth and other planets is proposed. A feature of this version is that the system of linear equations obtained by the discrete ordinate method is solved using the block elimination method. This is an exact and computationally efficient method; moreover it is easy to implement. The computer program developed based on this method is about two times faster than the program used in the free DISORT package.

Difference schemes based on the support operator method for fluids dynamics problems in a collector containing gas hydrates

Abstract: Difference schemes based on the support operator method are considered as applied to fluid dynamics in underground collectors containing gas hydrate deposits. A system of mass and energy balances describing fluid dynamics in a porous medium containing gas hydrate deposits is given. A dissipative hydrate equation is derived that determines the thermodynamic evolution of the parameters of the system. It is shown that the jumps in specific volumes and internal energy occurring in phase transitions play a crucial role in the stability of the evolution of the system in the dissipation thermodynamic module of the system. A family of rotation-invariant difference schemes of the support operator method on unstructured meshes is constructed for numerical computations. The schemes are tested on a series of model problems. Their numerical solutions are presented.

Asymptotic expansions of solutions in a singularly perturbed model of virus evolution

Abstract: An initial boundary value problem for a singularly perturbed system of partial integro-differential equations involving two small parameters multiplying the derivatives is studied. The problem arises in a virus evolution model. An asymptotic solution of the problem is constructed by the Tikhonov-Vasil’eva method of boundary functions. The analytical results are compared with numerical ones.

The Resolvent Approach for the Wave Equation

Abstract: Mixed problem for the wave equation (in the cases of fixed ends and periodic conditions) with minimal requirements to the initial data are studied. An approach based on the contour integration of the resolvent for the operator generated by the corresponding spectral problem is developed. This approach makes it possible to obtain a classical solution without using the asymptotics for eigenvalues or any information on eigenfunctions.

Production model in the conditions of unstable demand taking into account the influence of trading infrastructure: Ergodicity and its application

Abstract: A production model with allowance for a working capital deficit and a restricted maximum possible sales volume is proposed and analyzed. The study is motivated by an attempt to analyze the problems of functioning of low competitive macroeconomic structures. The model is formalized in the form of a Bellman equation, for which a closed-form solution is found. The stochastic process of product stock variations is proved to be ergodic and its final probability distribution is found. Expressions for the average production load and the average product stock are found by analyzing the stochastic process. A system of model equations relating the model variables to official statistical parameters is derived. The model is identified using data from the Fiat and KAMAZ companies. The influence of the credit interest rate on the firm market value assessment and the production load level are analyzed using comparative statics methods.

Numerical analysis of soliton solutions of the modified Korteweg-de Vries-sine-Gordon equation

Abstract: Multisoliton solutions of the modified Korteweg-de Vries-sine-Gordon equation (mKdV-SG) are found numerically by applying the quasi-spectral Fourier method and the fourth-order Runge-Kutta method. The accuracy and features of the approach are determined as applied to problems with initial data in the form of various combinations of perturbed soliton distributions. Three-soliton solutions are obtained, and the generation of kinks, breathers, wobblers, perturbed kinks, and nonlinear oscillatory waves is studied.

Computation of shock waves in plasma

Abstract: A multitemperature code intended for the numerical solution of the multicomponent gas dynamics equations in problems with a high energy density in matter is described. The velocities of all components with nonzero masses are assumed to be identical. Together with transport of gases with a tabulated equation of state, the code can include electron heat conduction, radiative transfer, energy exchange between the components, and chemical reactions. The gasdynamic part is based on Godunov’s scheme and an efficient Riemann solver with an approximate local equation of state.

Mathematical modeling of quasi-one-dimensional hemodynamics

Abstract: New unpublished results of research conducted under the guidance of Prof. Favorskii at the Chair of Computational Methods of Moscow State University are presented. Justification and implementation of a quasi-one-dimensional approximation for mathematical modeling of hemodynamics are discussed.

Transformation of sine-Gordon solitons in models with variable coefficients and damping

Abstract: The dynamics of sine-Gordon solitons in the presence of an external force, damping, and a spatially modulated periodic potential is studied. Numerical methods are used to show the possibility of generating localized nonlinear waves of the soliton and breather types. Their evolution is investigated, and the dependences of the amplitude and the oscillation frequency on the parameters of the system are found.

Parallel technology for numerical modeling of fluid dynamics problems by high-accuracy algorithms

Abstract: A parallel computation technology for modeling fluid dynamics problems by finite-volume and finite-difference methods of high accuracy is presented. The development of an algorithm, the design of a software implementation, and the creation of parallel programs for computations on large-scale computing systems are considered. The presented parallel technology is based on a multilevel parallel model combining various types of parallelism: with shared and distributed memory and with multiple and single instruction streams to multiple data flows.

Eigenvalue transmission problems describing the propagation of TE and TM waves in two-layered inhomogeneous anisotropic cylindrical and planar waveguides

Abstract: Problems on the propagation of surface TE and TM waves in an inhomogeneous anisotropic two-layered planar or cylindrical magneto-dielectric waveguide are considered. The problem is reduced to the analysis of a Sturm-Liouville problem of a special kind with boundary conditions of the third kind, nonlinearly depending on the spectral parameter. The conditions under which TE and TM waves can propagate are obtained, and the regions of localization of the corresponding propagation constants are determined.