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25References
1. Ivanov M.I., Kremer I.A., Laevsky Yu.M. A computational model of fluid filtration in fractured porous media //
Siberian J. Num. Math. 2021. V. 24, N. 2. P. 145-166.
2. Ivanov, M.I., Kremer, I.A., Urev, M.V. Solving the Pure Neumann Problem by a Finite Element Method // Numer.
Analys. Appl. 2019. V. 12, N. 4. P. 359�371. https://doi.org/10.1134/S1995423919040049.
Schemes for solving filtration problem of a heat-conducting two-phase liquid in a porous medium
M. I. Ivanov1, I. A. Kremer1,2, Yu. M. Laevsky1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2Novosibirsk State University
Email:[email protected], [email protected], [email protected]
DOI 10.24412/cl-35065-2021-1-00-22
This work is a continuation of the study of the problem of the motion of a two-phase liquid in a porous
medium [1]. In addition, the dependence of the oil viscosity on the temperature is determined, and the energy
equation is included in the system of equations. In the framework of the single-temperature model, the energy
equation is reduced to the heat equation, which describes the conductive mechanism of heat propagation in a
in a porous structure and in a heat-conducting liquid, as well as the convective heat transfer by the filtration
flow. The heat equation is written in a mixed generalized formulation. By analogy with the IMPES scheme, the
convective term is considered on the explicit time layer, and the integration of the conductive term is carried
out using the implicit scheme. This approach to the numerical solution of the heat equation allows to save the
value of the integration step and reuse previously developed codes for filtration problems. The representation
of phase velocities in the form of components co-directed with the total flow, and oppositely directed compo-
nents [2] provides a strict balance of heat in the grid elements. The properties of the proposed algorithm are
discussed on the examples of numerical solutions of model problems.
This work was supported by the RSF (grant 19-11-00048).
References
1. Ivanov M. I., Kremer I. A., Laevsky Yu. M. On the streamline upwind scheme of solution to the filtration problem //
Siberian Electronic Mathematical Reports. 2019. V. 16. P. 757-776. DOI:10.33048/semi.2019.16.051.
2. Ivanov M.I., Kremer I.A., Laevsky Yu. M. Numerical model of gravity segregation of two-phase fluid in porous
media based on hybrid upwinding // Russian J. of Numerical Analysis and Mathematical Modelling. 2021. V. 36, N 1.
P. 17-32. DOI: https://doi.org/10.1515/rnam-2021-0002.
Simulation of heat transfer with considering permafrost thawing in 3D media
D. A. Karavaev
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Email: [email protected]
DOI 10.24412/cl-35065-2021-1-00-23
An approach to mathematical modeling of heat transfer with permafrost algorithm [1, 2] in 3D based on
the idea of localizing the phase transition area is considered. The paper presents a problem statement for a
non-stationary heat transfer and a description of a numerical method based on a predictor-corrector scheme.
For a better understanding of the proposed splitting method the approximation accuracy was studied taking
into account inhomogeneous right-hand side. The phase changes in the numerical implementation of perma-
frost thawing is considered in the temperature range and requires recalculation of coefficients values of heat
equation at each iteration step in time. A brief description of parallel algorithm based on a three-dimensional
decomposition method and the parallel sweep method [3] is presented. A study of parallel algorithm imple-
mentations on the high-performance computing system of the Siberian Supercomputer Center of the SB RAS
was performed. The results of the permafrost algorithm work on models with one and several wells are also
presented.
This work was supported by the Russian Science Foundation (project 19-11-00048).
References
1 Vaganova N., Filimonov M. Simulation of freezing and thawing of soil in Arctic regions // IOP Conf. Series: Earth and
Environmental Science. 72. 012005. 2017. doi :10.1088/1755-1315/72/1/012005.
2. Stepanov S. P., Sirditov I. K., Vabishchevich P. N., Vasilyeva M. V., Vasilyev V. I., Tceeva A. N. Numerical Simulation
of Heat Transfer of the Pile Foundations with Permafrost // Lect. Notes in Computer Science. P. 625�632. 2017. doi:
10.1007/978-3-319-57099-0 71.
3. Federov A. A., Bykov A. N. Method of Two-Level Parallelization of the Thomas Algorithm for Solving Tridiagonal
Linear Systems on Hybrid Computers With Multicore Coprocessors // Numerical methods and programming. 2016. 17.
P. 234-244. doi: 10.26089/NumMet.v17r322.
The numerical modelling of satellite motion by the symmetric multi-step methods in the predictor-�orrector
mode
E. D. Karepova, I. R. Adaev, Yu. V. Shan�ko
Institute of Computational Modeling of SBRAS
Email: [email protected]
DOI 10.24412/cl-35065-2021-1-00-24
The orbital motion is described by the system of second-order ordinary differential equations which nu-
merical integration by the Stormer-Cowell multistep methods leads to a longitude error which increases quad-
ratically in time. This presents a problem when performing long-time integration. J. Lambert and I. Watson
proposed the symmetric methods [1], that possess a periodicity property when the product of the step-size
and the angular frequency lies within a certain interval called the interval of periodicity. The numerical integra-
tion of orbit by the symmetric methods with the step-size from the interval of periodicity gives the longitude
error which increases linearly, whereas the energy error remains roughly constant.
The symmetric methods are not uniquely determined even if their order and explicitness are specified.
We construct and investigate the high-order symmetric explicit and implicit methods in the "Predict-Evaluate-
Correct-Evaluate" (PECE) mode. The methods were selected according to size of the stability interval P(EC)kE
mode, the value of an error constant, behavior of the roots of the stability polynomial, and an accuracy of the
numerical solution of test problems.
This work was supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of science and high-
er education of the Russian Federation in the framework of the establishment and development of regional centers for
mathematics research and education (Agreement No. 075-02-2020-1631).
References
1. J. D. Lambert, I. A. Watson. Symmetric Multistep Methods for Periodic Initial Value Problems // J. Inst. Maths
Applics. 1976. V. 18. P. 189�202.
