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516 Plenary session
domains of solution smoothness. A characteristic feature of these algorithms is the use of two types of unknown
quantities -conservative variables related to the centers of the computational cells, and flow variables
related to the faces of the computational cells. In this case, flux variables are expressed not only in terms of
conservative values in adjacent cells, but also depend on flux variables from the previous time layer.
The conservative-characteristic algorithms are based on the finite volume method, and the flows on the
faces of the computational cells are calculated using the characteristic form of the equations. In each computational
cell, local Riemann invariants are constructed, the values of which on a new layer in time are calculated
by extrapolation or interpolation. Extrapolation algorithms include the CABARET scheme.
Conservative-characteristic algorithms have the second order of approximation on irregular computational
grids, which transforms into the first in the vicinity of strong discontinuities. The monotonicity of the solution
is achieved by nonlinear correction of fluxes based on the maximum principle, which does not depend on
any tuning parameters. As applied to the equations of gas dynamics, CH schemes make it possible to calculate
both shock-wave processes of any intensity and turbulent flows with incomplete resolution of the spectrum of
turbulent pulsations without tuning parameters, in particular, the generation of sound by turbulent jet and its
propagation in the near and middle zones.
The report will provide a brief overview of the use of CH algorithms in the problems of aeroacoustics, hydrogen
safety and computational oceanology.
On preconditioning of grid SLAEs using graph transformations
V. P. Il'in1,2,3, G. A. Omarova1,2, D. V. Perevozkin1,3, A. V. Petukhov1
1
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2
Novosibirsk State University
3
Novosibirsk State Technical University
Email: ilin@sscc.ru
DOI 10.24412/cl-35065-2021-1-00-19
The general principles and some specific implementations of the construction of graph preconditioners
[1, 4] for iterative methods in Krylov subspaces [2] are considered, focused on the fast solution of systems of
linear algebraic equations (SLAE) with sparse matrices of large orders that arise when approximating multidimensional
boundary value problems of mathematical modeling by the methods of finite differences, finite volumes,
finite elements and discontinuous Galerkin [6] algorithms of various orders of accuracy on unstructured
grids. Approaches to the construction of spanning trees for multidimensional weighted connected graphs are
investigated, a reduction algorithm for fast solution of grid equations on graphs is described, as well as issues
of parallelization of the proposed preconditioned iterative processes in Krylov subspaces. Variants of domain
decomposition and multigrid methods [3, 5] are considered as special cases of the proposed approach. The
results of numerical experiments are discussed that demonstrate the effectiveness of these algorithms on a
representative series of typical methodological examples.
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. Saad, Y. Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, 2003.
3. A. Napov and Y. Notay, An efficient multigrid method for graph laplacian system II: robust aggregation, SIAM J. SCI.
COMPUT., 39 (2017), No. 5, pp. S379-S403.
PLenary session 17
4. M. Bern, J. Gilbert, B. Hendrickson, N Nguyen, S. Toledo, Support-graph preconditioners, SIAM J. MATRIX ANAL.
APPL. V. 27, No. 4, pp. 930-951, 2006, Society for Industrial and Applied Mathematics.
5. K. D. Gremban, Combinatorial preconditioners for sparse, symmetric, diagonally dominant linear systems, School
of Computer Science Carnegie Mellon University Pittsburgh, PA 15213, 1996.
6. V.P. Il�in, Mathematical Modeling. Part I. Continuous and Discrete Models. Novosibirsk: SBRAS Publ., 2017.
Programmed cell death regulation using rationally designed molecular probes
N. V. Ivanisenko
Institute of Cytology and Genetics SB RAS
Email: ivanisenko@bionet.nsc.ru
DOI 10.24412/cl-35065-2021-1-02-44
There are two types of apoptosis induction: intrinsic-mediated via mitochondria and extrinsic-mediated
via death receptor (DR) activation. CD95/Fas is one of the most studied members of the DR family. The induction
of apoptosis via CD95 is largely controlled by the Death-Inducing Signaling Complex (DISC), which is
formed upon CD95 stimulation. The major components of the DISC complex include CD95, FADD, procaspases-
8/10 and c-FLIP (cellular FLICE inhibitory protein) proteins. Deregulation of the CD95 pathway accompanies a
variety of tumors and neurodegenerative diseases. Structural modeling of the key components of the DISC
complex and in silico screening of compounds targeting them have a great potential towards design of new
therapeutics and providing deep insights into molecular mechanisms of the signaling pathway functioning and
pathology development.
In the current study we applied structural modeling and virtual screening techniques of large databases of
chemical compounds to target the caspase-8/c-FLIPL complex. Designed chemical probe FLIPinB. was able to
target the heterodimerization interface leading to allosteric activation of the pro-apoptotic activity of the
complex. Kinetic mathematical model was further developed to analyze the observed effects of FLIPinB. on
DISC activation. Based on the modeling results we could predict that the stabilized FLIPinB./caspase-8/c-FLIPL
complex plays a major role at the very initial stages of the DISC assembly and procaspase-8 processing. Furthermore,
conducted structural analysis of the DISC complex suggests high therapeutic potential of c-FLIP targeting
compounds to enhance cell death in cancer cell lines that are characterized by high c-FLIP levels.
This work was supported by the Kurchatov Genomics Center of the Institute of Cytology & Genetics SB RAS (project
number: � 075-15-2019-1662).
Mesh conservation laws in filtration problems with discontinues solutions
M. I. Ivanov1, I. A. Kremer1,2, Yu. M. Laevsky1,2
1
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
2
Novosibirsk State University
Email: laev@labchem.sscc.ru
DOI 10.24412/cl-35065-2021-1-00-21
We proposed new efficient computational algorithms for solving a number of filtration problems for a
two-phase fluid in porous and fractured-porous media. Main feature of the considered problems is in the
presence of discontinuous solutions which prohibits consideration of the mathematical model in the form of a
system of differential equations in the entire computational domain. We have considered a model in the form
of integral laws of mass conservation and momentum in arbitrary subdomains. In particular, we formulated
Darcy's law in a generalized form for the total velocity, and the phase velocities are given by the product of the
total velocity with some functions that are discontinuous in general [1]. The resulting problem formulation
