Pub Date : 2024-04-01DOI: 10.1134/s0965542524020040
A. S. Demidov, A. S. Samokhin
Abstract
The paper presents explicit numerically implementable formulas for the Poincaré–Steklov operators, such as the Dirichlet–Neumann, Dirichlet–Robin, Robin1–Robin2, and Grinberg–Mayergoiz operators, related to the two-dimensional Laplace equation. These formulas are based on the lemma about a univalent isometric mapping of a closed analytic curve onto a circle. Numerical results for domains with very complex geometries were obtained for several test harmonic functions for the Dirichlet–Neumann and Dirichlet–Robin operators.
{"title":"Explicit Numerically Implementable Formulas for Poincaré–Steklov Operators","authors":"A. S. Demidov, A. S. Samokhin","doi":"10.1134/s0965542524020040","DOIUrl":"https://doi.org/10.1134/s0965542524020040","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper presents explicit numerically implementable formulas for the Poincaré–Steklov operators, such as the Dirichlet–Neumann, Dirichlet–Robin, Robin1–Robin2, and Grinberg–Mayergoiz operators, related to the two-dimensional Laplace equation. These formulas are based on the lemma about a univalent isometric mapping of a closed analytic curve onto a circle. Numerical results for domains with very complex geometries were obtained for several test harmonic functions for the Dirichlet–Neumann and Dirichlet–Robin operators.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-01DOI: 10.1134/s0965542524020064
N. B. Konyukhova
Abstract
For an autonomous system of (N) nonlinear ordinary differential equations considered on a semi-infinite interval ({{T}_{0}} leqslant t < infty ) and having a (pseudo)hyperbolic equilibrium point, the paper considers an (n)-dimensional ((0 < n < N)) stable solution manifold, or a manifold of conditional Lyapunov stability, which, for each sufficiently large (t), exists in the phase space of the system’s variables in the neighborhood of its saddle point. A smooth separatrix saddle surface for such a system is described by solving a singular Lyapunov-type problem for a system of quasilinear first-order partial differential equations with degeneracy in the initial data. An application of the results to the correct formulation of boundary conditions at infinity and their transfer to the end point for an autonomous system of nonlinear equations is given, and the use of this approach in some applied problems is indicated.
AbstractFor an autonomous system of (N) nonlinear ordinary differential equations considered on a semiinfinite interval ({{T}_{0}} leqslant t < infty ) and having a (pseudo)hyperbolic equilibrium point, the paper considers an (n)-dimensional ((0 <. n < N) stable solution manifold, or a manifold conditional Lyapunov stability, which, for each sufficient large (t) exist in the phase space；n < N）稳定解流形，或者说条件 Lyapunov 稳定流形，对于每个足够大的（t），该流形存在于系统鞍点附近的变量相空间中。通过求解初始数据具有退化性的准线性一阶偏微分方程系统的奇异 Lyapunov 型问题，描述了这种系统的光滑分离矩阵鞍面。文中给出了这些结果在无穷远处边界条件的正确表述及其向自治非线性方程系统终点的转移方面的应用，并指出了这种方法在一些应用问题中的应用。
{"title":"Smooth Lyapunov Manifolds for Autonomous Systems of Nonlinear Ordinary Differential Equations and Their Application to Solving Singular Boundary Value Problems","authors":"N. B. Konyukhova","doi":"10.1134/s0965542524020064","DOIUrl":"https://doi.org/10.1134/s0965542524020064","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>For an autonomous system of <span>(N)</span> nonlinear ordinary differential equations considered on a semi-infinite interval <span>({{T}_{0}} leqslant t < infty )</span> and having a (pseudo)hyperbolic equilibrium point, the paper considers an <span>(n)</span>-dimensional <span>((0 < n < N))</span> stable solution manifold, or a manifold of conditional Lyapunov stability, which, for each sufficiently large <span>(t)</span>, exists in the phase space of the system’s variables in the neighborhood of its saddle point. A smooth separatrix saddle surface for such a system is described by solving a singular Lyapunov-type problem for a system of quasilinear first-order partial differential equations with degeneracy in the initial data. An application of the results to the correct formulation of boundary conditions at infinity and their transfer to the end point for an autonomous system of nonlinear equations is given, and the use of this approach in some applied problems is indicated.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-01DOI: 10.1134/s0965542524020143
A. Zyl’, N. Zamarashkin
Abstract
The paper proves estimates of (varepsilon )-ranks for TT decompositions of tensors obtained by tensorizing the values of a regular function of one complex variable on a uniform square grid in the complex plane. A relation between the approximation accuracy and the geometry of the domain of regularity of the function is established.
{"title":"Estimation of QTT Ranks of Regular Functions on a Uniform Square Grid","authors":"A. Zyl’, N. Zamarashkin","doi":"10.1134/s0965542524020143","DOIUrl":"https://doi.org/10.1134/s0965542524020143","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper proves estimates of <span>(varepsilon )</span>-ranks for TT decompositions of tensors obtained by tensorizing the values of a regular function of one complex variable on a uniform square grid in the complex plane. A relation between the approximation accuracy and the geometry of the domain of regularity of the function is established.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-01DOI: 10.1134/s0965542524020039
L. T. Ashchepkov
Abstract
A method for constructing a feedback that ensures the attraction of trajectories of an affine system to an equilibrium state and to a given manifold is proposed. The feedback is found in an analytical form as a solution to an auxiliary optimal control problem. Sufficient conditions for the existence of the optimal control are given. Application of the proposed method to some classes of linear and nonlinear systems is discussed.
{"title":"Synthesis of an Optimal Stable Affine System","authors":"L. T. Ashchepkov","doi":"10.1134/s0965542524020039","DOIUrl":"https://doi.org/10.1134/s0965542524020039","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A method for constructing a feedback that ensures the attraction of trajectories of an affine system to an equilibrium state and to a given manifold is proposed. The feedback is found in an analytical form as a solution to an auxiliary optimal control problem. Sufficient conditions for the existence of the optimal control are given. Application of the proposed method to some classes of linear and nonlinear systems is discussed.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-21DOI: 10.1134/s0965542524010020
A. A. Belov, V. S. Khokhlachev
Abstract
Evaluation of one-dimensional integrals arises in many problems in physics and technology. This is most often done using simple quadratures of midpoints, trapezoids and Simpson on a uniform grid. For integrals of periodic functions over the full period, the convergence of these quadratures drastically accelerates and depends on the number of grid steps according to an exponential law. In this paper, new asymptotically accurate estimates of the error of such quadratures are obtained. They take into account the location and multiplicity of the poles of the integrand in the complex plane. A generalization of these estimates is constructed for the case when there is no a priori information about the poles of the integrand. An error extrapolation procedure is described that drastically accelerates the convergence of quadratures.
{"title":"Improving the Accuracy of Exponentially Converging Quadratures","authors":"A. A. Belov, V. S. Khokhlachev","doi":"10.1134/s0965542524010020","DOIUrl":"https://doi.org/10.1134/s0965542524010020","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Evaluation of one-dimensional integrals arises in many problems in physics and technology. This is most often done using simple quadratures of midpoints, trapezoids and Simpson on a uniform grid. For integrals of periodic functions over the full period, the convergence of these quadratures drastically accelerates and depends on the number of grid steps according to an exponential law. In this paper, new asymptotically accurate estimates of the error of such quadratures are obtained. They take into account the location and multiplicity of the poles of the integrand in the complex plane. A generalization of these estimates is constructed for the case when there is no a priori information about the poles of the integrand. An error extrapolation procedure is described that drastically accelerates the convergence of quadratures.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-21DOI: 10.1134/s0965542524010123
Yu. A. Poveshchenko, A. Yu. Krukovskii, V. O. Podryga, P. I. Rahimly
Abstract
An approach for describing the metric properties of a difference mesh for discretizing repeated rotational operations of vector analysis as applied to modeling electromagnetic fields is proposed. Based on the support operator method, integral-consistent operations (gradient, divergence and curl) are constructed, which are necessary to obtain estimates of the convergence of difference schemes for repeated rotational operations designed to solve specific problems of magnetohydrodynamics. Using smooth solutions of a model magnetostatic problem with first-order accuracy, the convergence of the difference schemes constructed in this work with a zero eigenvalue of the spectral problem is proved. In this case, no restrictions are imposed on the difference tetrahedral mesh, except for its nondegeneracy. Calculation of electromagnetic fields for a three-dimensional problem of magnetic hydrodynamics in a two-temperature approximation with the full set of spatial components of velocity and electromagnetic fields is presented. The dynamics of electromagnetic fields is developed against the background of rotational diffusion of the magnetic field vector.
{"title":"Convergence of Some Difference Schemes of the Support Operator Method for Repeated Rotational Operations","authors":"Yu. A. Poveshchenko, A. Yu. Krukovskii, V. O. Podryga, P. I. Rahimly","doi":"10.1134/s0965542524010123","DOIUrl":"https://doi.org/10.1134/s0965542524010123","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>An approach for describing the metric properties of a difference mesh for discretizing repeated rotational operations of vector analysis as applied to modeling electromagnetic fields is proposed. Based on the support operator method, integral-consistent operations (gradient, divergence and curl) are constructed, which are necessary to obtain estimates of the convergence of difference schemes for repeated rotational operations designed to solve specific problems of magnetohydrodynamics. Using smooth solutions of a model magnetostatic problem with first-order accuracy, the convergence of the difference schemes constructed in this work with a zero eigenvalue of the spectral problem is proved. In this case, no restrictions are imposed on the difference tetrahedral mesh, except for its nondegeneracy. Calculation of electromagnetic fields for a three-dimensional problem of magnetic hydrodynamics in a two-temperature approximation with the full set of spatial components of velocity and electromagnetic fields is presented. The dynamics of electromagnetic fields is developed against the background of rotational diffusion of the magnetic field vector.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-21DOI: 10.1134/s0965542524010068
L. S. Bryndin, B. V. Semisalov, V. A. Beliaev, V. P. Shapeev
Abstract
One-dimensional flows of an incompressible viscoelastic polymer fluid that are qualitatively similar to the solutions of Burgers’ equation are described on the basis of mesoscopic approach for the first time. The corresponding initial boundary-value problem is posed for the system of quasilinear differential equations. The numerical algorithm for solving it is designed and verified. The algorithm uses the explicit fifth-order scheme to approximate unknown functions with respect to time variable and the rational barycentric interpolations with respect to space variable. A method for localization of singular points of the solution in the complex plain and for adaptation of the spatial grid to them is implemented using the Chebyshev-Padé approximations. Two regimes of evolution of the solution to the problem are discovered and characterized while using the algorithm: regime 1—a smooth solution exists in a sufficiently large time interval (the singular point moves parallel to the real axis in the complex plane); regime 2—the smooth solution blows up at the beginning of evolution (the singular point reaches the segment of the real axis where the problem is posed). We study the influence of the rheological parameters of fluid on the realizability of these regimes and on the length of time interval where the smooth solution exists. The obtained results are important for the analysis of laminar-turbulent transitions in viscoelastic polymer continua.
{"title":"Numerical Analysis of the Blow-Up of One-Dimensional Polymer Fluid Flow with a Front","authors":"L. S. Bryndin, B. V. Semisalov, V. A. Beliaev, V. P. Shapeev","doi":"10.1134/s0965542524010068","DOIUrl":"https://doi.org/10.1134/s0965542524010068","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>One-dimensional flows of an incompressible viscoelastic polymer fluid that are qualitatively similar to the solutions of Burgers’ equation are described on the basis of mesoscopic approach for the first time. The corresponding initial boundary-value problem is posed for the system of quasilinear differential equations. The numerical algorithm for solving it is designed and verified. The algorithm uses the explicit fifth-order scheme to approximate unknown functions with respect to time variable and the rational barycentric interpolations with respect to space variable. A method for localization of singular points of the solution in the complex plain and for adaptation of the spatial grid to them is implemented using the Chebyshev-Padé approximations. Two regimes of evolution of the solution to the problem are discovered and characterized while using the algorithm: regime 1—a smooth solution exists in a sufficiently large time interval (the singular point moves parallel to the real axis in the complex plane); regime 2—the smooth solution blows up at the beginning of evolution (the singular point reaches the segment of the real axis where the problem is posed). We study the influence of the rheological parameters of fluid on the realizability of these regimes and on the length of time interval where the smooth solution exists. The obtained results are important for the analysis of laminar-turbulent transitions in viscoelastic polymer continua.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140205362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-21DOI: 10.1134/s0965542524010093
L. A. Kalyakin, E. G. Ekomasov
Abstract
A partial differential equation modeling the motion of a domain wall taking into account external magnetic fields and damping is considered. In the case of constant coefficients, this equation has a set of trivial solutions—equilibria. Solutions in the form of simple (traveling) waves that correspond to a dynamic transition from one equilibrium to another are studied. Possible types of waves that are stable in linear approximation are listed. A method for calculating the velocity of such waves is given.
{"title":"Simulation of Domain Walls: Simple Waves in the Magnetodynamics Equation","authors":"L. A. Kalyakin, E. G. Ekomasov","doi":"10.1134/s0965542524010093","DOIUrl":"https://doi.org/10.1134/s0965542524010093","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A partial differential equation modeling the motion of a domain wall taking into account external magnetic fields and damping is considered. In the case of constant coefficients, this equation has a set of trivial solutions—equilibria. Solutions in the form of simple (traveling) waves that correspond to a dynamic transition from one equilibrium to another are studied. Possible types of waves that are stable in linear approximation are listed. A method for calculating the velocity of such waves is given.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-21DOI: 10.1134/s0965542524010056
R. V. Brizitskii, A. A. Donchak
Abstract
The paper studies a multiplicative control problem for the reaction–diffusion equation in which the reaction coefficient nonlinearly depends on the substance concentration, as well as on spatial variables. The role of multiplicative controls is played by the coefficients of diffusion and mass transfer. The solvability of the extremum problem is proved, and optimality systems are derived for a specific reaction coefficient. Based on the analysis of these systems, the relay property of multiplicative and distributed controls is established, and estimates of the local stability of optimal solutions to small perturbations of both the quality functionals and one of the given functions of the boundary value problem are derived.
{"title":"Multiplicative Control Problem for a Nonlinear Reaction–Diffusion Model","authors":"R. V. Brizitskii, A. A. Donchak","doi":"10.1134/s0965542524010056","DOIUrl":"https://doi.org/10.1134/s0965542524010056","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper studies a multiplicative control problem for the reaction–diffusion equation in which the reaction coefficient nonlinearly depends on the substance concentration, as well as on spatial variables. The role of multiplicative controls is played by the coefficients of diffusion and mass transfer. The solvability of the extremum problem is proved, and optimality systems are derived for a specific reaction coefficient. Based on the analysis of these systems, the relay property of multiplicative and distributed controls is established, and estimates of the local stability of optimal solutions to small perturbations of both the quality functionals and one of the given functions of the boundary value problem are derived.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-21DOI: 10.1134/s0965542524010044
M. D. Bragin
Abstract
A new test problem for one-dimensional gas dynamics equations is considered. Initial data in the problem is a periodic smooth wave. Shock waves are formed in the gas flow over a finite time. The convergence under mesh refinement is analyzed for two third-order accurate linear schemes, namely, a bicompact scheme and Rusanov’s scheme. It is demonstrated that both schemes have only the first order of integral convergence in the shock influence area. However, when applied to equations of isentropic gas dynamics, the schemes converge with at least the second order.
{"title":"Actual Accuracy of Linear Schemes of High-Order Approximation in Gasdynamic Simulations","authors":"M. D. Bragin","doi":"10.1134/s0965542524010044","DOIUrl":"https://doi.org/10.1134/s0965542524010044","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A new test problem for one-dimensional gas dynamics equations is considered. Initial data in the problem is a periodic smooth wave. Shock waves are formed in the gas flow over a finite time. The convergence under mesh refinement is analyzed for two third-order accurate linear schemes, namely, a bicompact scheme and Rusanov’s scheme. It is demonstrated that both schemes have only the first order of integral convergence in the shock influence area. However, when applied to equations of isentropic gas dynamics, the schemes converge with at least the second order.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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