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518 Plenary session
naturally leads to a spatial approximation by the mixed finite element method for calculating the total velocity
and pressure, and the so-called centered finite volume method for calculating phase saturations. The time approximation
of the saturation equation is based on the explicit upwind Euler scheme. We developed this approach
both for one-porous and two-porous models describing flow in fractured-porous media using the mass
transfer function between pore blocks and fractures. In the case of a two-porous model, we built the upwind
scheme both within each medium and for the implementation of mass transfer between the media.
We considered the problem of gravitational segregation of a two-phase fluid in a porous medium [2]. We
constructed a new computational scheme for solving the multidimensional problem of gravitational segregation
of a two-phase fluid in a porous medium. Unlike the currently well-known IHU (Implicit Hybrid Upwinding)
approach proposed in [3], we have developed an explicit version of this approach (EHU) which is not inferior to
IHU in accuracy and significantly exceeds IHU in performance. Otherwise, the algorithm with the standard upwind
approximation is unable to adequately reproduce the filtration process. For the method we have constructed,
we constructed a proof of the weak maximum principle with an explicit indication of the Courant-
Friedrichs-Levy condition that ensures the stability and monotonicity of the scheme. In this case, for the Buck-
ley-Leverett model where the saturation dynamics is described by a hyperbolic equation, the obtained conditions
are not restrictive in terms of the time step size, and the time step limiting factor is accuracy only. This
ensures the competitiveness of the EHU vs. IHU for the specified set of problems.
This work was supported by the Russian Science Foundation (grant 19-11-00048).
References
1. Ivanov �. I., Kremer I. A., Laevsky Yu. M. Oil reservoir simulation based on the conservation laws in integral form
// AIP Conference Proceedings. 2020. V. 2312. P. 050008.
2. Ivanov �. 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 Num. Analysis and Math. Modelling. 2021. V. 36, Is.1. P. 17-32.
3. Lee S. H, Efendiev Y., Tchelepi H. A. Hybrid upwind discretization of nonlinear two-phase flow with gravity // Advances
in Water Resources. 2015. V. 82. P. 27-38.
3D inverse problems of magnetic susceptibility restoration from experimental data
I. I. Kolotov1, D. V. Lukyanenko1, Yanfei Wang2, A. G. Yagola1
1
Lomonosov Moscow State University
2
Institute of Geology and Geophysics, Chinese Academy of Sciences
Email: yagola@physics.msu.ru
DOI 10.24412/cl-35065-2021-1-01-98
Retrieval of magnetic parameters using magnetic tensor gradient measurements receives attention in recent
years. Traditional magnetic inversion is based on the total magnetic intensity data and solving the corresponding
mathematical physical model. In recent years, with the development of the advanced technology,
acquisition of the full tensor gradient data becomes available. In work [1], the problem of restoring magnetization
parameters has been solved. In this problem three scalar functions(components of the magnetization vector)
were recovered using data by five scalar functions(independent components of the magnetic tensor). In
our work [2] we consider the problem of magnetic susceptibility restoration using magnetic tensor gradient
measurements. In this work we have recovered one scalar function (magnetic susceptibility) using data by five
scalar functions(components of the magnetic tensor). As we are dealing with the physically overdetermined
problem we expect to receive better results than if the problem was just physically determined. At the current
PLenary session 19
moment, we have provided testing calculations using simulated data. Now we are testing our approach on
real data.
References:
1. Y. Wang, D. Lukyanenko, A. Yagola. Magnetic parameters inversion method with full tensor gradient data //
Inverse Problem and Imaging. 2019. V. 13, no. 4. P. 745-754, DOI. 10.3934/ipi.2019034.
2. Y. Wang, I. Kolotov, D. Lukyanenko, A. Yagola. Reconstruction of Magnetic Susceptibility Using Full
Magnetic Gradient Data June 2020 Computational Mathematics and Mathematical Physics 60(6):1000-1007,
DOI: 10.1134/S096554252006010X.
A diffusion-convection problem with a fractional derivative along the trajectory of motion
A. V. Lapin1, V. V. Shaidurov2
1
Sechenov University, Moscow
2
Institute of Computational Modeling, Siberian branch of RAS, Krasnoyarsk
Email: avlapine@mail.ru
DOI 10.24412/cl-35065-2021-1-00-36
A new mathematical model of the diffusion-convective process with "memory along the flow path" is proposed.
This process is described by a homogeneous one-dimensional Dirichlet problem with a fractional derivative
along the characteristic curve of the convection operator, or, in other words, with fractional material derivative.
A finite-difference scheme is constructed using an analogue of the well-known L1-approximation of
time-fractional derivative for the fractional material derivative and the conventional approximation of the diffusion
term.. The unique solvability of the constructed mesh scheme is proved. The stability estimates are derived
in the uniform mesh norm, and the accuracy estimates are given under the assumptions of sufficient
smoothness of the initial data and the solution of the differential problem. The presented results are based on
the article [1].
This work was supported by the Russian Scientific Foundation (grant 20-61-46017).
References
1. Lapin A. V., Shaidurov V.V. A diffusion-convection problem with a fractional derivative along the trajectory of
motion// RJNAMM. 2021. V.36, N.3. P. 157-163.
Numerically statistical investigation of efficacy of SEIR model
G. Z. Lotova1,2, V. L. Lukinov1,2, M. A. Marchenko1,2, G. A. Mikhailov1, D. D. Smirnov1
1
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2
Novosibirsk State University
Email: lot@osmf.sscc.ru
DOI 10.24412/cl-35065-2021-1-00-82
A comparative analysis of the differential and the corresponding stochastic Poisson SEIR-models [1, 2] was
performed for the testing problem of the epidemic COVID-19 in Novosibirsk modeling in the period from the
23rd of March 2020 to the 21th of June 2020, with the initial population N = 2 798 170 [3]. By varying the initial
population in the form N = n � m with m . 2, it was shown that the average values of the sick identified was
less (beginning with the 7th of April 2020) the corresponding differential values by the quantity that is statistically
not distinguished from C(t)/m, with C . 27.3 on the 21th June. This relationship allows to use the stochas-
