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Improved Iterative Methods for the Solution Grid Equations: Theory and...
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Improved Iterative Methods for the Solution Grid Equations: Theory and Application

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 13510

Special Issue Editors

Prof. Dr. Alexey Beskopylny

E-Mail Website
Guest Editor
Department of the Transport Systems, Don State Technical University, 344002 Rostov Oblast, Russia
Interests: mechanical properties; finite element analysis; structural analysis; structural dynamics; finite element modeling; mechanical behavior of materials; mechanical testing construction; construction engineering mechanics of materials
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Alexander Sukhinov

E-Mail Website
Guest Editor
Department of Mathematic and Informatic, Don State Technical University, Gagarin, 1, 344000 Rostov-on-Don, Russia
Interests: difference schemes; finite elements; methods for solving grid equations; numerical methods

Special Issue Information

Dear Colleagues,

This Special Issue aims to publish original research articles in constructing and investigating advanced difference schemes with improved dispersion and accuracy for diffusion–convection–reaction problems, which have arisen in modeling hydrophysical and hydrobiological processes for sea and coastal systems as well as for the Korteweg de Vries equation. The set of improved difference schemes, here presented and investigated, is based on linear combinations of leaf-frog and central difference schemes for relatively small grid Peclet numbers (less than number 2) and linear combinations of leaf-frog schemes and upwind schemes for large values of grid Peclet numbers. The original splitting schemes—two-dimensional–one-dimensional additive schemes—have been elaborated for convection–diffusion problems in natural systems. For the numerical solution of appropriate grid equations with non-self-adjoint operators, two variants of symmetric triangular–diagonal precondition methods have been built—one of variation type and the other using spectral estimations. The linearization on the time grid and convergence to the primary nonlinear task solutions of linearized boundary value problems has been investigated in L1 and L2 and its well-posedness. Additionally, investigations of related problems have been discussed—interpolation bottom boundary surfaces based on hyperbolic exponent splines. 

Potential topics include but are not limited to the following:

  1. Construction and study of the leaf-frog ("cabaret") difference scheme with improved dispersion properties for the Korteweg de Vries equation;
  2. Construction and study of the difference scheme leaf-frog ("cabaret") difference scheme with improved dispersion properties for the convection–diffusion equations;
  3. Optimization of the schemes with weights for the numerical solution of the convection–diffusion equation;
  4. Interpolation of reliefs and physical fields based on hyperbolic splines;
  5. Construction and study of locally-two-dimensional–locally-one-dimensional schemes for convection–diffusion problems in natural systems;
  6. Investigation of the convergence in L2 solutions of linearized on time grid chain boundary value problems for biogeochemical cycles to the origin nonlinear boundary value problem;
  7. An improved iterative alternating–triangular method for solving the convection–diffusion grid equations with a bounded grid Peclet number based on a priori spectral estimates;
  8. Adaptive iterative alternating–triangular method of variational type for solving the grid equations of convection–diffusion with a bounded grid Peclet number.

Prof. Dr. Alexey Beskopylny
Prof. Dr. Alexander Sukhinov
Guest Editors

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Keywords

  • diffusion–convection problems
  • difference schemes
  • splitting schemes
  • dispersion
  • accuracy
  • boundary value problems
  • quasi-linear parabolic equations
  • linearization
  • convergence in spaces L1, L2
  • grid equations
  • non-self-adjoint operators
  • grid Peclet number
  • iterative symmetric triangular methods

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Published Papers (8 papers)

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Research

15 pages, 320 KiB  
Article
Research of the Solutions Proximity of Linearized and Nonlinear Problems of the Biogeochemical Process Dynamics in Coastal Systems
by Alexander Sukhinov, Yulia Belova, Natalia Panasenko and Valentina Sidoryakina
Mathematics 2023, 11(3), 575; https://doi.org/10.3390/math11030575 - 21 Jan 2023
Cited by 2 | Viewed by 1120
Abstract
The article considers a non-stationary three-dimensional spatial mathematical model of biological kinetics and geochemical processes with nonlinear coefficients and source functions. Often, the step of analytical study in models of this kind is skipped. The purpose of this work is to fill this [...] Read more.
The article considers a non-stationary three-dimensional spatial mathematical model of biological kinetics and geochemical processes with nonlinear coefficients and source functions. Often, the step of analytical study in models of this kind is skipped. The purpose of this work is to fill this gap, which will allow for the application of numerical modeling methods to a model of biogeochemical cycles and a computational experiment that adequately reflects reality. For this model, an initial-boundary value problem is posed and its linearization is carried out; for all the desired functions, their final spatial distributions for the previous time step are used. As a result, a chain of initial-boundary value problems is obtained, connected by initial–final data at each step of the time grid. To obtain inequalities that guarantee the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems, the energy method, Gauss’s theorem, Green’s formula, and Poincaré’s inequality are used. The scientific novelty of this work lies in the proof of the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems in the norm of the Hilbert space L2 as the time step τ tends to zero at the rate O(τ). Full article
(This article belongs to the Special Issue Improved Iterative Methods for the Solution Grid Equations: Theory and Application)
21 pages, 1502 KiB  
Article
Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems
by Alexander Sukhinov, Alexander Chistyakov, Inna Kuznetsova, Yulia Belova and Elena Rahimbaeva
Mathematics 2022, 10(19), 3564; https://doi.org/10.3390/math10193564 - 29 Sep 2022
Cited by 4 | Viewed by 1331
Abstract
Modeling complex hydrodynamic processes in coastal systems is an important problem of mathematical modeling that cannot be solved analytically. The approximation of convective terms is difficult from the point of view of error reduction. This paper proposes a difference scheme based on a [...] Read more.
Modeling complex hydrodynamic processes in coastal systems is an important problem of mathematical modeling that cannot be solved analytically. The approximation of convective terms is difficult from the point of view of error reduction. This paper proposes a difference scheme based on a linear combination of the Upwind Leapfrog scheme with 2/3 weight coefficient, and the Standard Leapfrog scheme with 1/3 weight coefficient. The weight coefficients are obtained as a result of solving the problem of minimizing the approximation error. Numerical experiments show the advantage of the developed scheme in comparison with other modifications of the Upwind Leapfrog scheme in the case when the convective transport prevails over the diffusion one. The proposed difference scheme solves transport problems more effectively than classical difference schemes in the case when the Péclet number falls in the range from 2 to 20. It follows that the considered difference scheme allows hydrodynamic problems to be solved in regions of complex shape effectively. Full article
(This article belongs to the Special Issue Improved Iterative Methods for the Solution Grid Equations: Theory and Application)
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18 pages, 4493 KiB  
Article
Solving Hydrodynamic Problems Based on a Modified Upwind Leapfrog Scheme in Areas with Complex Geometry
by Alexander Sukhinov, Alexander Chistyakov, Inna Kuznetsova, Yulia Belova and Elena Rahimbaeva
Mathematics 2022, 10(18), 3248; https://doi.org/10.3390/math10183248 - 7 Sep 2022
Cited by 5 | Viewed by 1636
Abstract
In recent years, the number of adverse and dangerous natural and anthropogenic phenomena has increased in coastal zones around the world. The development of mathematical modeling methods allows us to increase the accuracy of the study of hydrodynamic processes and the prediction of [...] Read more.
In recent years, the number of adverse and dangerous natural and anthropogenic phenomena has increased in coastal zones around the world. The development of mathematical modeling methods allows us to increase the accuracy of the study of hydrodynamic processes and the prediction of extreme events. This article discusses the application of the modified Upwind Leapfrog scheme to the numerical solution of hydrodynamics and convection–diffusion problems. To improve the accuracy of solving the tasks in the field of complex shapes, the method of filling cells is used. Numerical experiments have been carried out to simulate the flow of a viscous liquid and the transfer of substances using a linear combination of Upwind and Standard Leapfrog difference schemes. It is shown that the application of the methods proposed in the article allows us to reduce the approximation error in comparison with standard schemes in the case of large grid numbers of Péclet and to increase the smoothness of the solution accuracy at the boundary. The soil dumping and suspended matter propagation problems are solved using the developed schemes. Full article
(This article belongs to the Special Issue Improved Iterative Methods for the Solution Grid Equations: Theory and Application)
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15 pages, 2146 KiB  
Article
The Construction and Research of the Modified “Upwind Leapfrog” Difference Scheme with Improved Dispersion Properties for the Korteweg–de Vries Equation
by Alexander Sukhinov, Alexander Chistyakov, Elena Timofeeva, Alla Nikitina and Yulia Belova
Mathematics 2022, 10(16), 2922; https://doi.org/10.3390/math10162922 - 13 Aug 2022
Cited by 4 | Viewed by 1458
Abstract
This paper covers the construction and research of a scheme to solve the problem with nonlinear dispersion wave equations, described by the model Korteweg–de Vries equation. The article proposes approximating the equation based on improved “Upwind Leapfrog” schemes. Its difference operator is a [...] Read more.
This paper covers the construction and research of a scheme to solve the problem with nonlinear dispersion wave equations, described by the model Korteweg–de Vries equation. The article proposes approximating the equation based on improved “Upwind Leapfrog” schemes. Its difference operator is a linear combination of operators of the “Standard Leapfrog” and “Upwind Leapfrog” difference schemes, while the modified scheme is obtained from schemes with optimal weight coefficients. Combining certain values of the weight coefficients mutually compensates for approximation errors. In addition, the modified scheme acquires better properties compared with the original schemes. The results of test calculations of solutions of the nonlinear Korteweg–de Vries equation are presented, illustrating the advantages of the modified scheme. Full article
(This article belongs to the Special Issue Improved Iterative Methods for the Solution Grid Equations: Theory and Application)
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17 pages, 1418 KiB  
Article
Mathematical Model of Suspended Particles Transport in the Estuary Area, Taking into Account the Aquatic Environment Movement
by Alexander Sukhinov, Alexander Chistyakov, Inna Kuznetsova, Yulia Belova and Alla Nikitina
Mathematics 2022, 10(16), 2866; https://doi.org/10.3390/math10162866 - 11 Aug 2022
Viewed by 1548
Abstract
A large amount of contaminants enter marine systems with river runoff, so the purpose of the study is to model the transport of suspended particles in the estuary area. To describe hydrodynamic and hydrophysical processes, the mathematical model of the suspended particles transport [...] Read more.
A large amount of contaminants enter marine systems with river runoff, so the purpose of the study is to model the transport of suspended particles in the estuary area. To describe hydrodynamic and hydrophysical processes, the mathematical model of the suspended particles transport was used, supplemented by a three-dimensional mathematical model of hydrodynamics, used to calculate the fields of the aquatic environment movement velocity vector, and equation for calculating the variable density. The approximation of the equations for calculating the velocity field by spatial variables is based on the splitting schemes for physical processes with fluid volume of the control areas, which allows for us to consider the complex geometry of the coastline and the bottom. The suspended particles transport model is approximated using splitting schemes for two-dimensional and one-dimensional problems. Numerical experiments were carried out to simulate the aquatic environment movement in the estuary area, the multicomponent suspension deposition, as well as mixing of waters in the mouth, taking into account the different density of the aquatic environment. The used models and methods allow to significantly improve the accuracy of modeling suspended particle transport and consider the factors influencing the studied processes. Full article
(This article belongs to the Special Issue Improved Iterative Methods for the Solution Grid Equations: Theory and Application)
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16 pages, 1259 KiB  
Article
Sufficient Conditions for the Existence and Uniqueness of the Solution of the Dynamics of Biogeochemical Cycles in Coastal Systems Problem
by Alexander Sukhinov, Yulia Belova, Alla Nikitina and Valentina Sidoryakina
Mathematics 2022, 10(12), 2092; https://doi.org/10.3390/math10122092 - 16 Jun 2022
Cited by 12 | Viewed by 1450
Abstract
The article considers a three-dimensional mathematical model of population dynamics based on a system of non-stationary parabolic advection-diffusion-reaction equations with lower derivatives describing the advective motion of the aquatic environment and non-linear source functions. In contrast to the previous authors’ works devoted to [...] Read more.
The article considers a three-dimensional mathematical model of population dynamics based on a system of non-stationary parabolic advection-diffusion-reaction equations with lower derivatives describing the advective motion of the aquatic environment and non-linear source functions. In contrast to the previous authors’ works devoted to the description of this model and its numerical implementation, this article presents the results of an analytical study of the initial-boundary value problem corresponding to this model. For these purposes, the original initial-boundary value problem is linearized on a single time grid—for all nonlinear sources, their final spatial distributions for the previous time step are used. As a result, a chain of initial-boundary value problems is obtained, connected by initial—final data at each step of the time grid. For this chain of linearized problems, the existence and uniqueness of the solution of the initial-boundary value problem for the system of partial differential equations in the Hilbert space were researched. Numerical experiments were performed for model problems of the main types of phytoplankton populations in coastal systems under the influence of dynamically changing biotic and abiotic factors, the results of which are consistent with real physical experiments. The developed model, including the proposed model of biological kinetics, allows for the study of the productive and destructive processes of the shallow water body biocenosis to assess the state of the processes of reproduction of valuable and commercial fish participating in the food chain with selected species of summer phytoplankton. Full article
(This article belongs to the Special Issue Improved Iterative Methods for the Solution Grid Equations: Theory and Application)
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11 pages, 296 KiB  
Article
Two-Dimensional-One-Dimensional Alternating Direction Schemes for Coastal Systems Convection-Diffusion Problems
by Alexander Sukhinov and Valentina Sidoryakina
Mathematics 2021, 9(24), 3267; https://doi.org/10.3390/math9243267 - 16 Dec 2021
Cited by 3 | Viewed by 1828
Abstract
The initial boundary value problem for the 3D convection-diffusion equation corresponding to the mathematical model of suspended matter transport in coastal marine systems and extended shallow water bodies is considered. Convective and diffusive transport operators in horizontal and vertical directions for this type [...] Read more.
The initial boundary value problem for the 3D convection-diffusion equation corresponding to the mathematical model of suspended matter transport in coastal marine systems and extended shallow water bodies is considered. Convective and diffusive transport operators in horizontal and vertical directions for this type of problem have significantly different physical and spectral properties. In connection with the above, a two-dimensional–one-dimensional splitting scheme has been built—a three-dimensional analog of the Peaceman–Rachford alternating direction scheme, which is suitable for the operational suspension spread prediction in coastal systems. The paper has proved the theorem of stability solution with respect to the initial data and functions of the right side, in the case of time-independent operators in special energy norms determined by one of the splitting scheme operators. The accuracy has been investigated, which, as in the case of the Peaceman–Rachford scheme, with the special definition of boundary conditions on a fractional time step, is the value of the second order in dependency of time and spatial steps. The use of this approach makes it possible to obtain parallel algorithms for solving grid convection-diffusion equations which are economical in the sense of total time of problem solution on multiprocessor systems, which includes time for arithmetic operations realization and the one required to carry of information exchange between processors. Full article
(This article belongs to the Special Issue Improved Iterative Methods for the Solution Grid Equations: Theory and Application)
16 pages, 3324 KiB  
Article
Mathematical Modeling of the Phytoplankton Populations Geographic Dynamics for Possible Scenarios of Changes in the Azov Sea Hydrological Regime
by Alexander Sukhinov, Yulia Belova, Alexander Chistyakov, Alexey Beskopylny and Besarion Meskhi
Mathematics 2021, 9(23), 3025; https://doi.org/10.3390/math9233025 - 25 Nov 2021
Cited by 8 | Viewed by 2039
Abstract
Increased influence of abiotic and anthropogenic factors on the ecological state of coastal systems leads to uncontrollable changes in the overall ecosystem. This paper considers the crucial problem of studying the effect of an increase in the water’s salinity in the Azov Sea [...] Read more.
Increased influence of abiotic and anthropogenic factors on the ecological state of coastal systems leads to uncontrollable changes in the overall ecosystem. This paper considers the crucial problem of studying the effect of an increase in the water’s salinity in the Azov Sea and the Taganrog Bay on hydrobiological processes. The main aim of the research is the diagnostic and predictive modeling of the geographic dynamics of the general phytoplankton populations. A mathematical model that describes the dynamics of three types of phytoplankton is proposed, considering the influence of salinity and nutrients on algae development. Discretization is carried out based on a linear combination of Upwind Leapfrog difference schemes and a central difference scheme, which makes it possible to increase the accuracy of solving the biological kinetics problem at large values of the grid Péclet number (Peh > 2). A software package has been developed that implements interrelated models of hydrodynamics and biogeochemical cycles. A modified alternating-triangular method was used to solve large-dimensional systems of linear algebraic equations (SLAE). Based on the scenario approach, several numerical experiments were carried out to simulate the dynamics of the main species of phytoplankton populations at different levels of water salinity in coastal systems. It is shown that with an increase in the salinity of waters, the habitats of phytoplankton populations shift, and marine species invasively replace freshwater species of algae. Full article
(This article belongs to the Special Issue Improved Iterative Methods for the Solution Grid Equations: Theory and Application)
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