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# Department of Computational Methods

Head of the department: Gulin Aleksey, Professor, Dr.Sc.

Since its foundation until February 2008 the Department was headed by the Hero of the Socialist Labor, the laureate of the Lenin and State Prizes, Academician of RAS Alexander Samarskii. The Department carries out research on fundamental problems of computational and applied mathematics, develops numerical methods for solving linear algebra problems and finite difference methods for studying boundary value problems of mathematical physics, and creates the related software.

Another direction of research is connected with mathematical modeling of complex applied problems in continuum mechanics, plasma physics, elasticity, chemical engineering, biophysics, ecology, medicine, sociology, as well as optical methods of information processing. There are laboratories of mathematical modeling in physics, of the difference methods and industrial mathematics at the Department.

### Staff members:

- Chetverushkin Boris, Academician of RAS, Professor, Dr.Sc.
- Popov Yury, Corresponding Member of RAS, Professor, Dr.Sc.
- Andreev Vladimir, Professor, Dr.Sc.
- Goloviznin Vasiliy, Professor, Dr.Sc.
- Elenin Georgy, Professor, Dr.Sc., Full Member of RNAN
- Mazhorova Olga, Professor, Dr.Sc.
- Mukhin Sergey, Professor, Dr.Sc.
- Sosnin Nikolay, Professor, Dr.Sc.
- Tishkin Vladimir, Professor, Dr.Sc.
- Trofimov Vyacheslav, Professor, Dr.Sc., Head of the laboratory of Mathematical modeling in physics
- Bogomolov Sergey, Professor, Dr.Sc., Deputy Director of the Kazakh Branch of MSU
- Abakumov Michail, Associate Professor, PhD
- Bunicheva Anna, Associate Professor, PhD, Scientific Secretary of the Department
- Voloshin Sergey, Associate Professor, PhD
- Esikova Natalia, Associate Professor, PhD
- Ilyutko Viktor, Associate Professor, PhD
- Ionkin Nikolay, Associate Professor, PhD
- Ternovsky Vladimir, Associate Professor, PhD
- Khapaev Michail, Associate Professor, PhD
- Khrulenko Alexander, Associate Professor, PhD
- Mikhailov Alexander, Associate Professor, Dr.Sc.
- Moiseev Tikhon, Senior Research Fellow, Dr.Sc.
- Mokin Andrey, Assistant Professor, PhD
- Isakov Victor, Assistant Professor, PhD
- Shlyakhov Pavel, Researcher
- Khapaeva Tatiana, Engineer

### Regular courses:

- Introduction into numerical methods by Prof. Gulin, 32 lecture hours, 3rd semester.
- Numerical methods by Prof. Andreev, 32 lecture hours, 5th semester.
- Numerical methods by Prof. Andreev, 32 lecture hours and 32 seminar hours, 6th semester.
- Methods of solving grid equations by Dr. Nikolaev, 32 lecture hours and 32 seminar hours, 5th and 6th semesters.
- Vector and tensor models, 32 lecture hours, 6th semester.
- Numerical methods of mathematical physics by Assoc. Prof. Abakumov, 64 lecture hours, 7th semester.
- Mathematical models of fluid dynamics by Prof. Mukhin, 64 lecture hours, 7th semester.
- Mathematical modeling in physics by Prof. Tishkin and Prof. Goloviznin, 32 lecture hours, 8th semester.
- Parallel Computing by Prof. Chetverushkin, 32 lecture hours, 9th semester.
- The Finite Element Method by Prof. Andreev, 32 lecture hours, 9th semester.

### Special courses:

- Theory of difference schemes stability by Prof. Gulin, Assoc. Prof. Ionkin, and Assoc. Prof. Morozova, one year.
- Numerical methods: non-standard approaches, half a year.
- Kinetic equations and particle method by Assoc. Prof. Bogomolov, half a year.
- Stochastic micro-macro modeling by Assoc. Prof. Bogomolov, half a year.
- Computational methods for gas dynamics by Prof. Popov, one year.
- Selected problems of classical and quantum mechanics and computational methods for their solution by Prof. Elenin, half a year.
- Computational methods for molecular dynamics by Prof. Elenin, half a year.
- Numerical Methods for Schroedinger-type equations by Prof. Trofimov, half a year.
- Mathematical modeling of the dynamic optical memory by Prof. Trofimov, half a year.
- Numerical methods in mathematical modeling by Dr. Savenkova, year and a half.
- Methods for the numerical solution of problems with moving boundaries by Prof. Mazhorova, half a year.
- Models of Perturbations in physics by Dr. Dorodnitsyn, half a year.
- Mathematical model EDA by Assoc. Prof. Khapaev, half a year.
- Mathematical models in hemodynamics by Prof. Mukhin, half a year.
- Numerical methods for hemodynamic modeling, half a year.
- Solving the linear hemodynamic equations by Prof. Sosnin, half a year.

### Special scientific seminars:

- Numerical methods for solving differential equations by Professors Gulin, Goloviznin, and Tishkin.
- Industrial mathematics and supercomputer simulations by Professors Goloviznin, Mukhin, Assoc. Professors Khapaev and Popova.
- Kinetic equations and parallel computing by Prof. Chetverushkin, Assoc. Prof. Bogomolov, and Dr. Dorodnitsyn.
- Mathematical modeling for problems of laser nanotechnology by Prof. Trofimov.
- Numerical simulation of electrochemical processes by Dr. Savenkova, Dr. Troshchiev, Dr. Shobukhov, and Dr. Ilyutko.
- Mathematical modeling in nanotechnology by Prof. Elenin and Dr. Shlyakhov.
- Modern problems of science and technology by Professors Popov, Mazhorova, Mukhin, and Assoc. Prof. Abakumov.

## Main Scientific Directions:

### Theory of numerical methods

(Professors V.B. Andreev, A.V. Gulin, G.G. Elenin, B.N. Chetverushkin, Y.P. Popov, A.P. Favorsky, S.V. Bogomolov, and Assoc. Prof. M.V. Abakumov).

The first area of research is difference schemes for the heat equation with nonlocal boundary conditions. Necessary and sufficient conditions for their stability with respect to initial data in the energy norm for double layer operator-difference schemes are found. A priori estimates that reflect the stability of difference schemes with respect to the right-hand sides are obtained. Difference schemes are constructed and studied for the heat equation with nonlocal boundary conditions with two real parameters. Estimates for the energy norm of the solution to the difference problem via the same norm of initial data are obtained. For difference schemes which are not stable in the usual sense the concept of asymptotic stability on the subspace is introduced and conditions of their asymptotic stability is given.

The second area of research deals with modeling different diffusion processes. For the singularly perturbed reaction-diffusion equation in a rectangle, a mixed boundary value problem is considered and its accuracy is evaluated. Estimates for the rate of convergence of difference schemes for the singularly perturbed convection-diffusion equations with a boundary layer are obtained. For the singularly perturbed stationary convection-diffusion equation in a rectangle with regular boundary layers on two adjacent sides, the monotonic finite-difference scheme on a Shishkin mesh is investigated. For the two-dimensional singularly perturbed convection-diffusion equation with constant coefficients in the half-plane with the boundary which is orthogonal to the direction of convection, anisotropic a priori estimates of the solution and its derivatives with the respect to the small parameter in the uniform metric are given under minimal assumptions on smoothness of the right-hand side of the equation and the boundary functions.

The third area investigates the conservatism property of two-staged symmetrically-symplectic techniques in the Runge-Kutta methods. An almost everywhere conservative symmetrically-symplectic computational method for solving the Cauchy problem for Hamiltonian systems is proposed. A manifold of the various orders of approximation in the space of parameters of two- and three-staged symplectic techniques in Runge-Kutta methods is studied in detail. A family of two-staged adaptive symmetrically-symplectic conservative Runge-Kutta methods of the second and the fourth order of approximation is constructed.

A method of constructing difference schemes of the Godunov type for approximating the system of equations of viscous compressible gas in cylindrical coordinates is developed. Finite-difference schemes for calculating flows of viscous compressible gas in cylindrical coordinates of the first and higher orders of approximation are constructed using the Row-Osher method. Gas flows between two rotating cylinders, in binary stellar systems, and in other problems of gravitational gas dynamics are investigated numerically.

Finally, one of the directions of scientific activity of the Department is the study of new algorithms that are apt for high-performance multiprocessor computing systems, as well as for solving problems of continuous media mechanics. In the framework of these studies the further development is given to the original kinetic schemes which were based on the idea of a continuum as a totality, i.e. a large number of discrete particles (molecules). This direction is focused on the direct use of finite-difference models for the one-particle distribution function. Wide possibilities of the kinetic approach to the simulation of complex problems of gas dynamics are caused by an easy parallel implementation of the algorithm. This feature approves the need for continuing this research in other areas of application. See also:

- V.B.Andreev, Pointwise approximation of corner singularities for singularly perturbed elliptic problems with characteristic layers // Intern. J. Numer. Anal. and Modeling., vol. 7, no. 3, pp. 416-427, 2010.
- A.V.Gulin, Asymptotic stability of nonlocal difference schemes // J. Numer. and Appl. Math., no. 2, pp. 34-43, 2011.

### Mathematical modeling of accretion disk binary star system

(Prof. Y.P. Popov, Prof. S.I. Mukhin and Assoc. Prof. M.V. Abakumov).

Mathematical modeling of dynamic processes in a binary star system is an urgent task as the majority of currently known stars are members of multiple and, in particular, binaries systems. A binary star system is characterized by an exchange of gas between system’s components and by formation of gas disks orbiting the star which is called accretion. Processes in accretion disks have a significant impact on the evolution of stars and are often the source of the X-ray emission which is especially important for observations. However the possibility of observing stellar systems is still limited, thus the primary means of research is mathematical modeling.

The evolution of the accretion disk in a binary star system is simulated. The accretion disk is described by a system of nonlinear equations of gas dynamics which takes into account the force of gravity of both components, centrifugal and Coriolis forces.

In order to solve the corresponding problem two-dimensional and three-dimensional gas-dynamic calculations on non-uniform difference grids are carried out. It is shown that the characteristic of the accretion disks is the structure of a binary system containing a pair of spiral waves which forms an important mechanism for redistributing angular momentum in the medium disk.

### Integrated nonlocal modeling of the cardiovascular system

(Professors S.I. Mukhin, N.V. Sosnin, A.P. Favorsky, Assoc. Professors M.V. Abakumov, A.Ya. Bunicheva, and A.B. Khrulenko).

The goal of this project, which was started in collaboration with the Faculty of Basic Medicine of MSU, is to create a mathematical model, numerical methods and corresponding software for numerical simulations of the cardiovascular flow. For this purpose the cardiovascular system is associated with the graph of vessels (edges) and tissues (nodes). Each vessel is taken as a one-dimensional flexible pipe, which is oriented in 3D space and connected either with other vessels or with tissues. Diameters of vessels are not constant and depend upon a great number of physiological and physical parameters, such as pressure, coefficient of flexibility, gravitation, etc. The vessel can be taken as it is or as a group of similar vessels. Tissues are characterized by their volume, by their ability to produce or sorb a certain amount of blood, by the Darcy coefficient, etc. Series of models of heart with different complexity are considered. Pressure, velocity of blood, diameter of a vessel which are estimated at any point of cardiovascular graph are taken as a basic function to be computed as a result of numerical simulation.

From mathematical point of view, the problem is stated as a system of nonlinear partial differential equations of viscous fluid dynamics on a graph. This system is approximated with an explicit conservative finite-difference scheme with specially designed linking conditions on a graph. This extremely complex system of nonlinear equations is solved by iterations.

The study relates to hemodynamics in a human body as a whole under the influence of a periodically contracting heart. Attention is paid to the cross-influence of various organs (in particular, kidneys) on pressure in the cardiovascular system. The study also takes in to account the influence of different factors connected with abnormal deviations of functional characteristics of vessels on a system as a whole, as well as different ways of compensating the vessels’ deficiencies, e.g. their bypass. One of the ways of adopting this study into practice is to analyze the impact and transportation of pharmacological substances within the cardiovascular system. Consideration of external influences, e.g. vibration, on the functioning of the cardiovascular system, is also planned.

In order to numerically implement the above mentioned nonlinear mathematical model in a graph, a conservative homogeneous implicit finite-difference scheme has been constructed on a directed graph. A system of finite-difference equations is supplemented with discrete analogues of correlations which simulate the performance of organs corresponding to certain nodes. Since this system of finite-difference equations of hemodynamics represents a system of non-linear algebraic equations for values of functions in the dots of a discrete grid at a new time row, the Newton method is used for its solving.

Possible applications of this project’s results include the following:

- Investigation of complex processing of the cardiovascular system in a normal and pathological modes.
- Investigation of possibilities of the arterial pressure regulation.
- Verification of models for units of the cardiovascular system.
- Investigation of specific types of the hemodynamic flow through the basic elements of the cardiovascular system (different types of vessels, the heart etc.).
- Development of methods for mathematical modeling of pharmacological substances propagation through the cardiovascular system with respect to their influence on hemodynamic processes.

The software that provides mathematical modeling of hemodynamics on any graph of the cardiovascular system is a complex of programs named CVSS (cardiovascular simulating system). These software permits:

- to estimate the proper initial data (initial values of pressure, velocity, cross-sections etc.);
- to modify the properties and numerical values of graph elements in an interactive mode;
- to provide the online 3D visualization of calculations results;
- to plot and modify any new or existing 3D graph of the cardiovascular system;
- to simulate numerically several pathologic processes in the cardiovascular system.

### Recent publications:

• 2013

- Bunicheva A.Ya., Mukhin S.I., Sosnin N.V., Favorskii A.P., Menyailova M.A. Studying the influence of gravitational overloads on the parameters of blood flow in vessels of greater circulation // Math. Models and Computer Simulation. 2013. 5. N 1. P. 81-91.
- Gulin A.V. On the spectral stability in subspaces for difference schemes with nonlocal boundary conditions // Differential Equations. 2013. 49. N 7. P. 815-823.
- Moiseev Т.Е. On the solvability of the Tricomi problem with a generalized Frankl transmission condition // Differential Equations. 2013. 49. N 6. P. 785-789.

• 2012

- Borzov A.G., Mukhin S.I., Sosnin N.V. Conservative schemes of matter transport in a system of vessels closed by the heart // Differential Equations. 2012. 48. N 7. P. 919-928.
- Elenin G.G., Aleksandrov P.A. On the conservativeness of a two-parameter family of three-stage symmetric-symplectic Runge-Kutta methods // Differential Equations. 2012. 48. N 7. P. 965-974.
- Gulin A.V. Stability of nonlocal difference schemes in a subspace // Differential Equations. 2012. 48. N 7. P. 940-949.
- Gulin A.V., Chetverushkin B.N. Explicit schemes and numerical simulation using ultrahigh-performance computer systems // Doklady Mathematics. 2012. 86. N 2. P. 681-683.
- Kupriyanov M.Yu., Khapaev M.M., Divin Y.Y., Gubankov V.N. Anisotropic distributions of electrical currents in high-Tc grain-boundary junctions // JETP Letters. 2012. 95. N 6. P. 289-294.
- Laponin V.S., Savenkova N.P., Il'utko V.P. Numerical method for soliton solutions // Computational Mathematics and Modeling. 2012. 23. N 3. P. 254-265.
- Moiseev T.E. Effective integral represention of a boundary value problem with mixed boundary conditions // Doklady Mathematics. 2012. 85. N 3. P. 347-349.
- Moiseev T.Е. On the solution of the Gellerstedt problem for the Lavrent'ev-Bitsadze equation // Differential Equations. 2012. 48. N 10. P. 1-1.
- Mukhin S.I., Sosnin N.V., Borzov A.G. Conservative algorithm of substance transport over a closed graph of cardiovascular system // Russ. J. Numer. Anal. and Math. Modeling. 2012. 27. N 5. P. 413-429.
- Ternovskii V.V., Khapaev M.M. Data surfaces of physical experiments // Doklady Mathematics. 2012. 86. N 1. P. 537-538.
- Khapaev M., Kupriyanov M.Yu. Sparse approximation of fem matrix for sheet current integro-differential equation // Matrix Methods: Theory, Algorithms And Applications. N.Y.: Word Scientific Publ., 2012. P. 510-522.