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10tiscale 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.
References
1. Petrashen G. I. Solutions of vector limit problems of mathematical physics in the case of a sphere // Reports of the
academy of sciences. 1945. V. 46. N. 7 P. 291-294.
Preconditioning methods based on spanning tree algorithms and Schur complement techniques
G. A. Omarova1,2, D. V. Perevozkin1
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2Novosibirsk State University
Email: foxillys@gmail.com
DOI 10.24412/cl-35065-2021-1-00-42
This work continues the evaluation of tree-based preconditioners started in [1, 2]. The new approach sug-
gests using Schur complement [3] to invert the preconditioning matrix instead of using direct solvers for build-
ing the preconditioner. This allows to reduce both memory footprint and computation time. Also, this ap-
proach is a good candidate for parallelization. We compare its convergence rate and performance with some
well-known preconditioners and solvers. The comparison is performed using SLAEs from the University of Flor-
ida�s sparse matrix collection [4].
References
1. Pravin M. Vaidya. Solving linear equations with symmetric diagonally dominant matrices by constructing good
preconditioners. Unpublished manuscript. A talk based on the manuscript was presented at the IMA Workshop on Graph
Theory and Sparse Matrix Computation, October 1991, Minneapolis.
2. Chen D, Toledo S. Implementation and evaluation of Vaidya�s preconditioners. Preconditioning 2001. 2001.
3. Saad, Y. Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, 2003.
4. SuiteSparse Matrix Collection Formerly the University of Florida Sparse Matrix Collection. [Electron. resource].
URL: https://sparse.tamu.edu/ (date of access: 27.05.2021).
Comparative analysis of methods for solving SLAE in three-dimensional initial-boundary value problem
M. S. Pekhterev, V. P. Ilin, V. S. Gladkikh
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Email: maxim-pekhterev@mail.ru
DOI 10.24412/cl-35065-2021-1-00-43
An experimental study of the performance of rendering methods for systems of linear algebraic equations
(SLAE) in implicit approximation schemes for three-dimensional boundary value problems in modeling non-
stationary heat conduction processes with phase transitions is carried out [1, 4]. Algorithms of finite volumes
on an unstructured grid [2] and various approaches to temporal approximation [5] are considered. To improve
the performance of the preconditioned iterative processes used in the Krylov subspaces [3], the technique of
choosing the optimal initial residuals at each time step is used. The effectiveness of the proposed approaches
is demonstrated in a representative series of methodological experiments.
This work was supported by the RSF-19-11-00048.
References
1. Vabishchevich P. N., Samarsky A.A. Computational heat transfer // M. Editorial URSS, 2003.
2. Petukhov A.V. Solution of the three-dimensional complex Heimholtz equation by the method of barycentric finite
volumes // Avtometriya, publishing house SB RAS. 2007. T. 43. No. 2. 112�124.
