Modified Upwind Leapfrog difference scheme

Сухинов, А. И.; Chistyakov, A. E.; Kuznetsova, Inna; Protsenko, E.A.; Belova, Yulia

doi:10.23947/2587-8999-2020-1-1-56-70

Cited by 3 publications

(5 citation statements)

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“…The proposed difference scheme has an advantage over the classical Upwind and Standard Leapfrog schemes in the case of small Courant numbers. In [24], the approximation error of the developed scheme, which is equal to O ch 2 , is researched, while for classical schemes, the approximation errors are equal to O h 2 . However, for cases of interest from the point of view of applications, the constant c is much less than unity, which implies the advantage of the proposed difference scheme.…”

Section: Discussionmentioning

confidence: 99%

“…Based on Fourier analysis, the increase in the accuracy of the solution when using scheme ( 14) occurs due to the mutual compensation of errors in the "phase velocity" of the harmonics of the solutions obtained on the basis of schemes (3) and ( 4). The stability and dispersion properties of a linear combination of the Upwind and Standard Leapfrog difference schemes are discussed in detail in [23,24].…”

Section: Linear Combination Of Upwind and Standard Leapfrog Schemesmentioning

confidence: 99%

“…Next, we consider functions having a derivative of the order α satisfying the inequality f (α) (x) ≤ K, with a period of 2π. There is an estimation of the residual term of the series (24) for any natural α [31,32]:…”

Section: Accuracy Of Solving the Heat Conduction Problemmentioning

confidence: 99%

“…To solve this class of problems, a difference scheme was proposed, which is a linear combination of Standard Leapfrog and Upwind Leapfrog schemes [23]. The considered schemes are included in a linear combination with weight coefficients of 1/3 and 2/3, respectively, which are obtained from the solution of minimizing the order of the approximation error problem [24].…”

Section: Introductionmentioning

confidence: 99%

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Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

et al. 2022

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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.

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Section: Discussionmentioning

confidence: 99%

Section: Linear Combination Of Upwind and Standard Leapfrog Schemesmentioning

confidence: 99%

Section: Accuracy Of Solving the Heat Conduction Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

et al. 2022

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“…The paper considers algorithms and software modules that implement the solution of grid equations in areas with geometry "elongated" along one of the spatial directions" arising from the numerical solution of problems of hydrodynamics and hydrobiology [8].…”

Section: Introductionmentioning

confidence: 99%

A software package for solving grid equations in areas with geometry "stretched" along one of the spatial directions

Litvinov

Rudenko²,

Gracheva³

et al. 2022

E3S Web Conf.

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A method for solving grid equations in areas with geometry "stretched" along one of the spatial directions is proposed. Method proposed for model includes pollutants whose concentration in the Azov Sea, according to the analysis of scientific publications and environmental databases, exceeds the maximum permissible concentration (MPC). The mechanism of the preconditioner formation is described. The estimation of the minimum and maximum eigenvalues is performed. Using object-oriented analysis and design, a class library has been implemented to solve the problem of impurity propagation in a shallow reservoir. The architecture of an information system to support scientific research capable of functioning in conditions of using heterogeneous computing environments is described. The system is based on the requirement of the maximum possible use of free software in order to minimize capital and operational costs. The system uses classes that allow you to read and write information about the geometry of the object under study and the parameters of the calculated grid. Numerical experiments were carried out using the developed software package, including the proposed mathematical model, focused on the multiprocessor computing system (MCS). The dependences of the SLAE solution time on the number of diagonals and the order of the matrix are obtained. The number of iterations for solving grid equations has been experimentally determined depending on the ratio of the number of nodes in spatial coordinates.

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Modeling the Process of Mixing Water in the Estuarine Area in the Presence of a Density Gradient of the Aquatic Environment Using Hybrid MPI/OpenMP Technology

Сухинов

Atayan

Kuznetsova

et al. 2021

Communications in Computer and Information Science

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Modified Upwind Leapfrog difference scheme

Cited by 3 publications

References 0 publications

Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

A software package for solving grid equations in areas with geometry "stretched" along one of the spatial directions

Modeling the Process of Mixing Water in the Estuarine Area in the Presence of a Density Gradient of the Aquatic Environment Using Hybrid MPI/OpenMP Technology

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