# Mirela Vinerean-Bernhoff

## Forskning

My research interests are mainly in kinetic theory and related topics. I was mainly working with the discrete velocity models (DVMs) for the Boltzmann equation, i.e. models where the velocity variable (or a corresponding variable) only takes a finite number of values, for them. It is a well-known fact that DVMs can approximate the Boltzmann equation up to any order. In my work I considered the general problem of the construction of discrete kinetic models (DKMs) with given conservation laws. This problem was first stated in connection with discrete models of the Boltzmann equation (BE), when it became clear that the velocity discretization can lead to equations with spurious conservation laws. The problem has been addressed in the last decade by several authors. Even though a practical criterion for the non-existence of spurious conservation laws has been devised, and a method for enlarging existing physical models by new velocity points without adding non-physical invariants has been proposed, a general algorithm for the construction of all normal (physical) discrete models with assigned conservation laws, in any dimension and for any number of points, was still lacking in the literature. We developed such a general algorithm in our work. We introduced the most general class of discrete kinetic models and obtain a general method for the construction and classification of normal DKMs. In particular, it is proved that for any given dimension d ≥ 2 and for any sufficiently large number N of velocities there is just a finite number of distinct classes of DKMs. We applied the general method in the particular cases of discrete velocity models (DVMs) of the inelastic BE and elastic BE. Using our general approach to DKMs and our results on normal DVMs for a single gas, we developed a method for the construction of the most natural (from physical point of view) subclass of normal DVMs for binary gas mixtures (they have the property that by isolating the velocities of single gases involved in the mixture, we also obtain normal models). In a joint work with Niclas Bernhoff we introduced generalizations to DVMs for multiple-component mixtures. We also developed some general algorithms for constructing such models. We showed that we can construct a supernormal DVM for any given number of species and any given rational mass ratios.

Pedagogical projects:

• Project “Datorstöd i matematikkurser” (2007) in the course “Pedagogy for University” : I have investigated how the use of computers and computer programs affects the teaching of mathematics at university level. In particular, I was investigated this in the course Geometry (C level) using for the classical and analytical geometry the program “Cabri Geometre” and for the differential geometry the program Mathematica.

• Project “Geometry with project” (2009): I developed the D-level course in Geometry containing the evolution of geometry (Euclidean, Analytical, Vector geometry, Differential, Non-Euclidian) and completed with a project part that can be with the emphasis on mathematics or mathematical didactics. The course was a part of an in-service training (kompetensutveckling) of junior lecturers (adjunkter) in the department of mathematics Karlstad University, but also a PhD course for PhD students in Mathematics and Mathematics Didactics.

• Project ” Wallenbergs Minnesfond”(2010): project founded by Stiftelsen Marcus och Amalia Wallenbergs Minnesfond. The project will be done together with the high school “Älvkullen” Karlstad. The first step in the project is planned to be an in-service course for the teachers in mathematics from Älvkullen: they learn how to use the program Geogebra, and also in which way they should use it to make from it a successful tool in teaching mathematics. The second step is the implementation of the program in the courses of mathematics at the high school, with the assistance from the university. The third step is the evaluation of the project and research concerning the pedagogy that was used.

• Project "An exploratory approach to engineering mathematics using GeoGebra" (2015): project founded by Karlstads universitet utvecklingsmedel för Flexibel utbildning och Blended learning. We have idenitified a potential for change and modernization in a first-semester course in mathematics for Bachelor students in engineering. Our goals were to increase the awareness about the relevance of mathematics for the program, to stimulate in-depth learning, and to find efficient ways of communicating mathematical knowledge. Our project is in line with current research on higher education in mathematics. We have introduced partly flipped classroom using screencast lectures to offer flexible, available, learning resources and allow for other scheduled student-centered activities. We have used dynamic visualizations in order to support mathematical understanding. We also changed partly the examination by introducing GeoGebra lab works.

## Undervisning

Undervisar matematik kurser inom ingenjörsutbildningen och lärarutbildningen.

## Biografi

- Född 4 juli 1974 i Sibiu, Rumänien.
- Utbildnig: - ämneslärare i matematik och datavetenskap (1997 Rumänien);

- doktor i matematik (2005 Karlstad)

- Jobbar på Kau sedan 1 juli 2001 (doktorand i matematik 2001-2005 i FMB Uppsala, vikarierande universiteslektor 2006, postdok 2007-2010 i FMB Uppsala, universiteslektor sedan 2011).
- Gift; har två barn.

## Utvalda publikationer

A. V. Bobylev and M. C. Vinerean, *Discrete kinetic models and conservation laws*, in Modelling and Numerics of Kinetic Dissipative Systems, L. Pareschi and G. Russo and G. Toscani, eds., Nova Science Publishers, 2006, pp. 147-162.

A. V. Bobylev and M. C. Vinerean, *General Methods of the Construction of Discrete Kinetic Models with Given Conservation Laws*, Rarefied Gas Dynamics, M.S.Ivanov and A.K.Rebrov, eds., Publishing House of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 2007, pp. 209-214.

A. V. Bobylev and M. C. Vinerean, *Construction and Classification of Discrete Kinetic Models without Spurious Invariants*, Riv. Mat. Univ. Parma (7) 7 (2007), pp.1-80.

A. V. Bobylev and M. C. Vinerean, *Discrete kinetic models with given invariants and their construction*, J. Stat. Phys., 132 (2008), pp. 153-170.

A. Bobylev, M. Vinerean and Å. Windfäll, *Discrete velocity models of the Boltzmann Equation and conservation laws*, Kinet. Relat. Models, 3, no.1 (2010), pp. 35-58.

M. C. Vinerean, Å. Windfäll and A. V. Bobylev, *Construction of normal discrete velocity models of the Boltzmann equation*, Nuovo Cimento, 33 (2010), pp. 257-264.

A. V. Bobylev and M. C. Vinerean, *Symmetric extensions of normal discrete velocity models.* 28th International Symposium on Rarefied Gas Dynamics 2012, M. Mareschal and A. Santos, eds., AIP Conference Proceedings 1501, American Institute of Physics, 2012, 254-261.

N. Bernhoff and M. Vinerean, *Discrete Velocity Models for Mixtures without Nonphysical Collision Invariants, 2016 (submitted)*

## Publikationer

- Mats Brunström, Eva Mossberg, Mirela Vinerean, 2016
- Niclas Bernhoff, Mirela Vinerean Bernhoff, 2016
- Alexander Bobylev, Mirela Vinerean (Bernhoff), 2012
- Mirela Vinerean (Bernhoff), Åsa Windfäll, Alexander Bobylev, 2010
- Alexander Bobylev, Mirela Vinerean (Bernhoff), Åsa Windfäll, 2010
- Alexander Bobylev, Mirela Vinerean-Bernhoff, 2008
- Alexander Bobylev, Mirela Vinerean-Bernhoff, 2007
- Alexander Bobylev, Mirela Vinerean-Bernhoff, 2007
- Alexander Bobylev, Mirela Vinerean-Bernhoff, 2006
- Mirela Vinerean-Bernhoff, 2005
- Mirela Vinerean-Bernhoff, 2004