Alexander Alekseenko, Assistant Professor,
Department of Mathematics
California State University Northridge
18111 Nordhoff St.,
Northridge, CA 91330-8313
DETERMINISTIC SOLUTION OF THE BOLTZMANN EQUATION
I have developed deterministic solvers for the Boltzmann equation of gas dynamics based on DG discretization. The kinetic approach is a microscopic approach, with the meaning that kinetic description targets processes happening in gas on the level of molecules. The main benefit of the kinetic approach is that on the molecular level is it relatively easy to describe different kinds of physical, mechanical and chemical interactions. The kinetic Boltzmann equation gives relatively simple means to aggregate information from all interactions over all molecules. However, there is a price to pay. The convenience of kinetic description comes at the expense of high dimensionality. In particular, if N is the number of degrees of freedom in one spatial dimension and in one velocity dimension, then the computational time required for completing one temporal step grows as O(N^11). This is vry fast. Just to put is in perspective if for N=10 one temporal step takes 0.01 second on one processr. Then N=100 will require about 32 years on the same processor. This implies that only the most efficient methods may be used for the solution of the Boltzmann equation. Some of the publications related to the development of such methods can be found on the right.
I am interested in the design of Runge-Kutta discontinuous Galerkin methods for the solution of initial-boundary value problems for the Einstein equations. In particular, an interesting and am interested in methods for formulating equations and specifying boundary conditions in the way that achieves a stable evolution. I am studying ways to solve systems with differential constraints numerically. In particular, I am working on the development of the methods for prescribing well-posed boundary conditions in a manner consistent with the constraint equations and methods for constraint enforcement during the simulations such as constraint-damping, incorporating constraints into the evolution and design of the constraint-free discretization techniques.
FINITE ELEMENTS / DISCONTINUOUS GALERKIN METHODS
I am interested in the development of higher order discontinuous Galerkin methods for solving hyperbolic problems encountered in gas kinetics such as Boltzmann-BGK equation. I am also working with applications of DG methods for second-order hyperbolic equations and model problems of constraint evolution. Also the development of fully implicit high order schemes for kinetic equations is very interesting.
DIFFUSION WEIGHTED MRI
The diffusion weighted MRI can measure rate of diffusion in a sample medium at any point in any direction. This information can be used to study orientation of the tissue fibers in the medium, since the fastest diffusion is assumed to occur along the fibers in the sample. By combining diffusion information, structure of the entire object can be revealed. I work with a CSUN student, John Sikora, on a research projects related to the fiber structure reconstruction in human brain based on the diffusion weighted MRI.
My Ph.D. thesis was devoted to applications of optimal control theory to inverse problems of electromagnetic wave propagation. Problems of electromagnetic wave propagation arise, e.g., in non-invasive techniques of medical imaging, non-destructing testing of materials in engineering, radar sensing and geophysical prospecting. In these applications the information about the object can only be acquired on the surface of the object or on its outside layers. The problem then consists in determining the material parameters inside the object from the data measured on the surface. This translates into the mathematical (inverse) problem of reconstructing the coefficients (or the right side) in the governing differential equation from the information known on the boundary of the domain.
PAPERS AND PREPRINTS
with E. Josyula, Deterministic Solution of the Boltzmann Equation Using Discontinuous Galerkin Discretizations in Velocity Space. submitted to JCP. http://arxiv.org/abs/1301.1099
with E. Josyula, Deterministic solution of the Boltzmann equation using a discontinuous Galerkin velocity discretization. Proceedings of the 28th International Symposium on Rarefied Gas Dynamics, Spain 2012, AIP Conference Proceedings, 8 pp.
with S. Gimelshein and N. Gimelshein, The application of discontinuous Galerkin space and velocity discretization to model kinetic equations. to appear in International Journal of Computational Fluid Dynamics. (2011).
Numerical properties of constraint-preserving boundary conditions for a second orderwave equation. In preparation for resubmission
Numerical properties of high order discrete velocity solutions to the BGK kinetic equation. Applied Numerical Mathematics (2009). Vol. 61 (2011), pp. 410–427. http://dx.doi.org/10.1016/j.apnum.2010.11.005
with A. Alexeenko and C. Galitzine, High order discontinuous Galerkin method for Boltzmann model equations, In: Proceedings of 40th AIAA Thermophysics Conference, Seattle, WA, June 23--26, 2008.
Constraint-preserving boundary conditions for the linearized BSSN formulation.Abstract and Applied Analysis, Vol. 2008, Article ID 742040 (2008), http://www.hindawi.com/getarticle.aspx?doi=10.1155/2008/742040
Well-posed initial-boundary value problem for a constrained evolution system and radiation-controlling constraint-preserving boundary conditions. Journal of Hyperbolic Differential Equations, Vol.4 (2007), No.4, pp.587--612. http://www.worldscinet.com/jhde/04/0404/S02198916070404.html http://arXiv.org/abs/gr-qc/0611011
with D. Arnold, New first-order formulation for the Einstein equations, Physical Review D, vol.68 (2003), 064013. http://arXiv.org/abs/gr-qc/0210071
with S. Kabanikhin, Numerical algorithms for identification problem in magnetoencephalography, Journal of Inverse and Ill-Posed Problems, vol.7 (1999), No.5, pp.387-408
with S. Kabanikhin,Uniqueness of the stationary point for the one-dimensional inverse problem in two-dimensional space, Journal of Inverse and Ill-Posed Problems, vol.6 (1998), No.2, pp.95-114
Two examples of constraint evolution: 3-1=2 and 6-3=5. research notes, California State University, Northridge (2006)