History
The former Laboratory of Numerical Methods (presently Department of Computational
Mathematics) was created in 1987 by some members of the Department of Mathematical
Modeling. The aim was to further strengthen the direction of numerical methods for PDEs as the
most important component in mathematical modeling for industrial applications. Prof. Raytcho
Lazarov was appointed for its first Head. Further heads were Prof. Michail Kaschiev (1991-2004)
and Assoc. Prof. Natalia Kolkovska.
A number of applied projects were developed in collaboration with
 Institute of Metallurgy and Metal Sciences;
 Institute of Microelectronics;
 Technical University of Sofia;
 Joint Institute for Nuclear Research (Dubna, Russia);
 Institute of Mathematical Modeling of Russian Academy of Sciences;
 Texas A&M University;
 Darmstadt University of Technology (Germany);
 Institute of Science and Technology of University of Manchester;
 Engineering Department of Queen Mary College (University of London).
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15 PhD and a lot of MSc. students wrote their theses under the supervision of department
members.
The department was involved in the organization of 6 International conferences on Numerical
Methods and Applications in Sofia and Borovets.
Members of the department participated in Scientific, Program or Organizing committees of
more than 40 other conferences.
In the period 1988-2003 about 15 scientists from the department got regular positions or
PhD/PostDoc fellowships at:
 Institute for Parallel Processing;
 Sofia University;
 Texas A&M University (USA);
 University of Texas in Austin (USA);
 University of California, Los Angeles (USA);
 Penn State University (USA);
 Technical University of Eindhoven (The Netherlands);
 University of Nijmegen (The Netherlands);
 Fraunhofer Institut Techno- und Wirtschaftsmathematik in Kaiserslautern (Germany);
 etc.
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Research report for the period 2004-2008
Scientific staff:
1.
Prof. DSc. Raytcho Lazarov – retired in 2008, currently at Texas A&M University;
2.
Prof. DSc. Mihail Kaschiev - deceased 2007;
3.
Assoc. Prof. Dr. Natalia Kolkovska;
4.
Assoc. Prof. Dr. Oleg Iliev – currently at FhG ITWM, Kaiserslautern;
5.
Dr. Ivan Bazhlekov;
6.
Dr. Milena Dimova;
7.
Dr. Ivan Georgiev – PostDoc at RICAM, Linz;
8.
Dr. Stanislava Stoilova;
9.
Dr. Daniela Vasileva;
10. Dr. Ludmil Zikatanov – only in 2004, currently at Penn State University;
11. Polya Dobreva – part-time in 2004-2005, currently at Institute of Mechanics.
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Main fields of research
 Theoretical investigation and practical realization of numerical methods and algorithms
for PDEs, systems of PDEs and integral equations - construction and analysis for
 finite element, finite difference and finite volume approximations;
 multilevel and domain decomposition methods;
 a posteriori error control and adaptive grid refinement.
 Mathematical modeling and numerical simulation of physical, fluid dynamic, chemical,
electrostatic, thermodynamic, biomechanical problems, etc.

formation and ionization of hydrogen like atoms and ions in magnetic fields;

magnetosheath-magnetosphere 3D system;

filtration processes, non-Newtonian and multiphase flows in plain and porous media;
 drop dynamics (breakup and coalescence) in complex non-Newtonian and viscoelastic
multiphase flows in presence of surfactants;

elasticity problems;

glass crystallization processes;

effects of electrostatic surface forces;

formation of structures in nonlinear heat-transfer media.
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Formal and informal co-operation and relations
 within the Academy
 joint work and participation in scientific projects of Institute for Parallel Processing, BAS;
 joint work with Institute of Mechanics and Institute of Physical Chemistry, BAS;
 at national level
 participation in a scientific project of Technical University, Gabrovo;
 joint work with Faculty of Mathematics and Informatics, Faculty of Physics and Faculty of
Chemistry, Sofia University;
 in Europe and world wide
 scientific projects and joint work with JINR, Dubna, Russia;
 participation in scientific projects and joint work with Czech (Institute of Geonics), Polish,
Hungarian, Austrian (RICAM) Acad. Sci.;
 participation in scientific projects of EC FP5, EC FP6;
 joint work with FhG-ITWM, Kaiserslautern (Germany), CWI, Amsterdam (The
Netherlands), Technical University of Eindhoven (The Netherlands), Texas A&M University
(USA).
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Most important data




number of scientific papers published in journals abroad: 67 (listed in SCI expanded: 54);
number of scientific papers published in Bulgarian journals: 3;
number of scientific papers published in conference proceedings: 18;
number of scientific reports (published by FhG-ITWM, RICAM, TAMU, JINR, etc.): 18;
 number of citations appeared in the period 2004-2008: more than 500, most cited: Prof. R.
Lazarov: more than 300 citations.
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 participation in teaching at Sofia University and South-West University of Blagoevgrad:
 lectures on Mathematics, Calculus, Numerical methods;
 seminars on Applied mathematics, Mathematical modeling, Numerical methods.
 post-graduate training at BAS: Theory of approximations.
 co-organization (with FMI, SU and IPP, BAS) of
Sixth International Conference on Numerical Methods and Applications,
August 20-24, 2006, Borovets, Bulgaria:
 128 participants, 70 from abroad;
 116 lecturers, 68 from abroad.
 organization of regular seminars on Computational mathematics.
 participation in editorial boards:
 Computational Methods in Applied Mathematics (Prof. R. Lazarov, Assoc.Prof. O. Iliev);
 East-West Journal on Numerical Mathematics (Prof. R. Lazarov);
 Numerical Methods for Partial Differential Equations (Prof. R. Lazarov);
 Mathematical Modelling and Analysis (Assoc.Prof. O. Iliev).
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 participation in councils, commissions and other expert bodies of external for BAS
institutions:
 Council on Informatics and Mathematical modeling, National Supreme Commission for
Attestation, 2004-2007 (Prof. M. Kaschiev);
 Council of the Faculty of Mathematics and Natural Sciences, South-West University,
Blagoevgrad, 2002-2006 (Prof. M. Kaschiev);
 Expert Commission for Scientific Cooperation with JINR-Dubna, 2000-2007 (Prof. M.
Kaschiev);
 Working Group WG 2.5, International Federation for Information Processing, 1988present (Prof. R. Lazarov);
 International Society for Porous Media, 2008-present (Assoc.Prof. O. Iliev, presidentelect 2009-).
 study/research visits:
 Dr. Ivan Bazhlekov, TUE, Eindhoven, The Netherlands, 01.01.2001-31.12.2004;
 Dr. Daniela Vasileva, CWI, Amsterdam, The Netherlands, 01.09.2003-31.05.2004;
 Prof. Michail Kaschiev, JINR, Dubna, Russia, 03.06.-27.06.04, 12.06.-01.07.2005;
 Dr. Stanislava Stoilova, JINR, Dubna, Russia, 16.06.-27.06.2004;
 Dr. Ivan Georgiev, Institute of Geonics, Ostrava, Czech Respublic, 25.06.-09.07.2005;
 Dr. Ivan Georgiev, RICAM, Linz, Austria, 03.10.-16.12.2005, 01.09.2008-31.08.2009.
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Scientific awards/recognition
 Prof. R. Lazarov - Doctor Honoris Causa of Sofia University, 2006;
 Dr. I. Georgiev - Award of Bulgarian Academy of Sciences for Young Scientists, 2006;
 Prof. R. Lazarov - Medal of Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences, 2008;
 Prof. R. Lazarov - Pichoridis Distinguished Lectureship, University of Crete, Greece, June 2008;
 Prof. R. Lazarov - Erasmus Mundus Visiting Scholar Award, University of Kaiserslautern, July
2008 - June 2011.
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A sample of scientific results
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Prof. Mihail Kaschiev, Dr. Milena Dimova (with S. Vinitsky et al. (JINR, Dubna))
A new efficient numerical method for
calculating the energy levels of lowlying exited states of a hydrogen atom
in a strong magnetic field is developed.
This method is based on the modern
implementation of the Kantorovich
approach to the parametric eigenvalue
problems in spherical coordinates. The
initial
two-dimentional
spectral
problem for the Schrödinger equation
is reduced to a one-dimentional
spectral parametric problem for the
angular variable and a finite set of
ordinary second-order differential
equations for the radial variable. The
resulting systems are solved using
high-order accuracy approximations of
the finite element method.
The approach elaborated provides a useful tool for
calculations of threshold phenomena in formation and
ionization of (anti)hydrogen like atoms and ions in
magnetic traps and channeling of ions in thin films.
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Assoc.Prof. Natalia Kolkovska, Dr. Ivan Georgiev (with Prof. I. Avramov (Inst. Physical
Chem.) and Prof. Chr. Russel (Fr.-Schiller Univ., Jena, Germany))
The crystal growth in multi-component systems with crystal composition different from that of the
ambient phase is simulated. The mathematical problem is a special case of moving boundary
problems. The boundary immobilization method is applied to solve numerically the diffusion
equations in an unknown region. A variety of physical characteristics, as concentration profiles, the
size of the growing crystal, are calculated for different physical parameters. Adequate interpretation of
the results is given.
Time dependence of the size of the growing crystal.
Concentration profiles at different times
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Assoc.Prof. Natalia Kolkovska
Numerical methods for solving second order elliptic problems with specific boundary condition,
given by a sum of normal derivative and a second order elliptic operator in tangential variables
are proposed and investigated. Optimal error estimates of the numerical methods in Sobolev
spaces are proved. Similar theoretical results are established for elliptic problems with
discontinuous coefficients and interface conditions of the same type.
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Assoc.Prof. Natalia Kolkovska, Dr. Daniela Vasileva (with Dr. R. Slavchov (Faculty of
Chemistry, Sofia University))
An algorithm for numerical simulation of surface forces, acting on AFM (atomic force
microscope) is developed. The mathematical model considers three phases (tip, water, dielectric)
and two interface surfaces – tip-water and water-dielectric. On each interface the surface dielectric
permittivities are modifying the conditions of the Gauss law. For this model a finite difference
method in cylindrical coordinates on a non-uniform grid, aligned with both interfaces, is
developed. The numerical experiments show that surface dielectric permittivities of tip-water and
water-dielectric give a strong addition to the image force pulling the AFM tip toward the dielectric
surface investigated.
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Assoc.Prof. Oleg Iliev, Dr. Daniela Vasileva (with Dr. D. Stoyanov (FhG-ITWM) and
Prof. W. Doerfler (Univ. Karlsruhe))
An adaptive refinement multigrid solver for
numerical simulation of flow of nonNewtonian fluid in saturated porous media
is developed. The mathematical model
consists of the continuity equation and the
generalized Darcy law. The numerical
method is based on a finite volume
discretization with mass conservation on the
interfaces between the coarser and finer
grids
and
second
order
accurate
discretisation for the fluxes.
Results from numerical solution of various
academic and practice-induced problems
demonstrate that the adaptive local
refinement approach allows to obtain the
same accuracy as in the full grid case, but
using significantly less memory and CPU
time.
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Assoc. Prof. Oleg Iliev, Dr. Daniela Vasileva
A local refinement algorithm for computer simulation of flow through oil filters is developed.
The mathematical model is based on laminar incompressible Navier-Stokes equations for the
flow in pure liquid zones and Brinkman extension to Darcy model for the flow in the porous
zone. A finite volume method on cell-centered locally refined grids is used for the discretization
and special attention is paid to the conservation of the mass on the interface between the coarse
and the fine grid.
A variety of numerical experiments
are performed and the results show
that the solver could be successfully
used for simulation of coupled flow
in plain and porous media. The local
refinement
ensures
a
significant
acceleration of the computations and
saving of memory, which is very
important in the case of 3D numerical
simulations.
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Dr. Ivan Bazhlekov (with Prof. H. Meijer and Dr. P. Anderson (TU Eindhoven))
A three-dimensional boundary
integral method for deformable
drops in viscous flows is
developed. The method is based
on a new nonsingular contourintegral representation. The
contour integration overcomes
the main difficulty with
boundary-integral calculations:
the singularities of the kernels.
It also improves the accuracy of
the calculations as well as the
numerical stability.
Drop deformation and breakup
in shear flow are shown in the
figure. Topological transition at
time t=53.6 is also shown.
Simulations
of
large
deformation and topological
transition are possible due to
the developed semi-automatic
adaptive mesh refinement.
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Dr. Ivan Bazhlekov (with Prof. A. Chesters; Prof. F. van de Vosse; Prof. H. Meijer and
Dr. P. Anderson (TU Eindhoven))
Mathematical
model
and
corresponding
numerical methods for simulation of 2D and 3D
drop coalescence in complex non-Newtonian and
viscoelastic multiphase flows are developed. In
2D case the problem is solved numerically by
means of a finite difference method for the
equations in the continuous phase and a boundary
integral method or finite-element method in the
drops. In 3D case the numerical method is based
on the nonsingular boundary integral method. In
this class of problems an important feature is the
presence of a thin film of thickness 3-5 orders of
magnitude smaller than the drop size. In order to
improve the resolution in the film zone a higherorder interface approximation is introduced.
Successfully
are
simulated
drop-to-drop
interaction (shown in the figure in the case of
external compressional flow) as well as foam
dynamics.
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Dr. Ivan Bazhlekov (with Prof. H. Meijer and Dr. P. Anderson (TU Eindhoven))
Numerical model for computer simulation of the effect of insoluble surfactants on drop dynamics is
developed. The mathematical model consists of: Stokes equation in the fluid phases; stress-balance
boundary condition on the interfaces; convection-diffusion equation on the evolving interface governs
the distribution of the surfactant concentration, which in turn determines the interfacial tension. The
numerical method is a combination of a three-dimensional boundary-integral method for the
hydrodynamics and a finite-volume method to solve the coupled fluid dynamics and surfactant
transport problem.
The model is applied successfully for 3D simulation of drop deformation and breakup (left figure),
drop-to-drop interaction (right figure) and foam dynamics. The color bar represents the surfactant
concentration.
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drop deformation and breakup (movie)
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drop-to-drop interaction (movie)
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foam dynamics (8 drops in a larger drop, movie)
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Dr. Milena Dimova (with Prof. S. Dimova (Faculty of Math.&Inf., Sofia University))
Self-similar solutions of a nonlinear heat-conduction equation
with a volume source under blow-up conditions are considered.
The self-similar problem is a BVP for a nonlinear elliptic
equation with nonunique solution. An efficient numerical
method for investigation of the eigenfunctions of the self-similar
problem is developed. This method is based on the Continuous
Analog of Newton Method and the Method of Finite Elements.
The completely new types of egenfunctions depending on the
values of the parameters of the medium and the choice of the
initial approximations are obtained
– two-dimensional
eigenfunctions with “zero” regions and “spiral” egenfunctions.
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Dr. Ivan Georgiev (with Prof. S. Margenov (IPP-BAS))
An algorithm for the numerical solution of the Lamé equations of elasticity in the case of mesh
anisotropy and coefficient jumps is developed. A preconditioned conjugate gradient (PCG) method
is applied for iterative solution of the linear algebraic system obtained after non-conforming finite
element discretization. Displacement decomposition of the stiffness matrix is used as a first step of
the algorithm. At the second step, modified incomplete factorization MIC(0) is applied to a proper
auxiliary M-matrix to get an approximate factorization of the obtained block-diagonal matrix.
Computer simulation of a pile foundation system in a multi-layer soil media: vertical strains;
vertical stresses, vertical displacements
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Dr. Ivan Georgiev (with Dr. J. Kraus (RICAM, AAS) and Prof. S. Margenov (IPP-BAS))
Algebraic multilevel iteration methods for three-dimensional elliptic problems discretized by a
family of Rannacher Turek non-conforming finite elements are developed. The derived estimates
of the constant  in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality allow the
efficient multilevel extension of the related two-level preconditioners. Representative numerical
tests well illustrate the optimal complexity of the resulting iterative solver, also for the case of
non-smooth coefficients.
128^3 voxels
256^3 voxels
μFEM simulation - microstructure analysis of bones
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Dr. Daniela Vasileva (with Prof. P.W.Hemker and A. Kuut (CWI, Amsterdam))
A multigrid adaptive mesh-refinement algorithm is developed for the solution of convectiondiffusion problems. The method is based on discontinuous Galerkin (Baumann-Oden DG)
discretization. The numerical experiments show that the algorithm may be successfully used for
resolution of boundary and interior layers.
Further an adaptive semirefinement technique is developed and the comparison with the adaptive
refinement algorithm shows that significantly less computer resources may be used for layers,
almost parallel to the x or y axis.
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Additional information about activities of the
department's members (CVs, list of publications,
etc.) may be found on the web-site of the
Institute:
http://www.math.bas.bg/new/site/
?call=USE~structure;&action=sing
le&id=10&lang=en
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