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10(m, k)-methods for DAEs of index 2
A. I. Levykin1,2, A. E. Novikov3, E. A. Novikov4
1Novosibirsk state University
2Institute of Computational Mathematics and Mathematical Geophysics SB RAS
3Siberian federal University, Krasnoyarsk
4Institute of computational modeling SB RAS
Emails: lai@osmf.sscc.ru, aenovikov@bk.ru, novikov@icm.krasn.ru
DOI 10.24412/cl-35065-2021-1-00-37
Authors derived non-iterative (m, k)-schemes for solving the Cauchy problem for differential-algebraic sys-
tems of index not exceeding 2. For these schemes, including freezing regularizing matrix, authors obtained and
studied accuracy and stability conditions. Formulas for transformation of (m, k)-schemes parameters are given.
L-stable (3, 2)-method of order 2 is derived. It requires two calls of a function, single evaluation of the Ja-
cobian matrix, and single LU-decomposition per integration step. The integration algorithm of alternating step
size is based on the new method. The algorithm allows to solve differential-algebraic equation systems of in-
dex not exceeding 2. Numerical results confirming the efficiency and reliability of new algorithm are given.
This work is partially performed within the framework of ICM&MG SB RAS state assignment (project 0251-2021-
0002). This work is 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. 075-02-2021-1388).
References
1. E. Hairer, G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems � Berlin :
Springer-Verlag, 1996. � 614 p.
2. Levykin, A.I., Novikov, A.E. & Novikov, E.A. Schemes of (m, k)-Type for Solving Differential-Algebraic and Stiff
Systems. Numer. Analys. Appl. 13, 34�44 (2020).
3. Novikov, A.E., Levykin A.I., Novikov E.A. (m, k)-Methods for Control Theory Problems.// 15th International Asian
School-Seminar Optimization Problems of Complex Systems, OPCS 2019, P. 120-124.
Non-conforming finite element modelling of coupled heat and mass transfer processes in phase-change
media
S. I. Markov1,2, E. P. Shurina1,2, N. B. Itkina1,3
1The Trofimuk Institute of Petroleum Geology and Geophysics SBRAS
2Novosibirsk State Technical University
3Institute of Computational Technologies SBRAS
Email: www.sim91@list.ru
DOI 10.24412/cl-35065-2021-1-00-39
Today, mathematical modelling is one of the ways to study natural phenomena and physical processes
taking place in the world around us. In heterogeneous media with phase-changing physical properties, math-
ematical modelling of coupled heat and mass transfer processes is complicated by the geometrical multiscale
of systems under study, the nonlinear dependence of physical fields, and the high contrast of the physical me-
dium properties. The formulated features determine the requirements for a computational scheme, which
should naturally preserve the regularity properties of mathematical models of physical processes at a discrete
level. For solving the problems in media with time-varying inter-fragments boundaries, non-conformal finite
element methods are most suitable. In the paper, we propose modified computational schemes of the mul-
tiscale discontinuous Galerkin method for approximating the system of Navier-Stokes-Darcy equations and
Stefan's problem. As a representative of the family of non-conforming methods, the discontinuous Galerkin
method provides freedom in the choice of function spaces and trace operators. The form of operators is de-
termined by the specifics of the problem being solved. To discretize physical fields, we use hierarchical bases
of the H(div) and H1 functional spaces. To solve finite element discrete analogues, algebraic multilevel solvers
are applied.
The research was supported by RSF (project No. 20-71-00134).
A conservative sixth-order algorithm for the direct Zakharov � Shabat problem
S. B. Medvedev1,2, I. A. Vaseva1,2, I. S. Chekhovskoy1, M. P. Fedoruk1,2
1Novosibirsk State University
2Federal research center for information and computational technologies SB RAS
Email: vaseva.irina@gmail.com
DOI 10.24412/cl-35065-2021-1-00-40
Improving the accuracy and efficiency of numerical algorithms for the direct Zakharov � Shabat (ZS) prob-
lem is an urgent problem in optics. We present a family of conservative sixth-order schemes for the ZS prob-
lem. The schemes are based on the generalized Cayley transform. In particular, we present an exponential
scheme similar to [1] and schemes based on rational approximation, which allowed the use of fast algorithms.
The schemes are compared with CF4[6] scheme [2]. Numerical experiments have shown the efficiency of the
new schemes.
The work of S.B.M. (analytics) was supported by the Russian Science Foundation (grant No.17-72-30006), the work of
I.A.V. and I.S.Ch. (numerical results) was supported by the state funding program FSUS-2020-0034.
References
1. Medvedev S., Vaseva I., Chekhovskoy I., Fedoruk M. Exponential fourth order schemes for direct Zakharov �
Shabat problem // Opt. Express. 2020. V. 28 (1), P. 20-39.
2. Chimmalgi S., Prins P. J., Wahls S. Fast nonlinear Fourier transform algorithms using higher order exponential
integrators // IEEE Access. 2019. V. 7, P. 145161-145176.
About one method for solving problems with quasispherical symmetry
V. V. Novikov, L. N. Fevralskikh
National Research Lobachevsky State University of Nizhni Novgorod
Email: grigorieva_ln@mail.ru
DOI 10.24412/cl-35065-2021-1-00-41
An approach based on the use of the apparatus of spherical vectors [1] is demonstrated for solving prob-
lems of mathematical physics with symmetry close to spherical. Several problems are shown, for which an ana-
lytical solution has been obtained. It helps to discover the qualitative features of the dynamics of the object
under study. The problem of free rotation of an elastic quasi-ball is considered. The possibility of global
movement of the axis of stable stationary rotation in the body is shown. The solution is obtained for the prob-
lem of the motion of a viscous fluid between non-concentric spherical and ellipsoidal surfaces. It was found
that the solution contains a radial flow. The possibility of generating a magnetic field by the found flow is in-
vestigated.
