I am a postdoctoral associate in the Department of Mathematics and Statistical Sciences at the University of Alberta, Canada.
My research interest is numerical analysis of partial differential equations modelling flow and magnetohydrodynamics problems.
PhD in Mathematics, 2014
University of Pittsburgh
M.S. in Mathematics, 2009
North Dakota State University
B.S. in Mathematics, 2007
National University of Uzbekistan
A. Takhirov, R. Frolov and P. Minev, Massively parallel direction splitting scheme for Navier-Stokes equations in spherical geometries, IN PROGRESS.
A. Takhirov and C. Trenchea, Second order, computationally efficient adaptive nonlinear filtering scheme for high Reynolds number flows, IN PROGRESS.
J.-L. Guermond, C. Nore and A. Takhirov, Numerical simulation of Madison dynamo experiment, IN PROGRESS.
A. Takhirov and J. Waters, Improved algorithm for parametrized flow problems with energy stable open boundary conditions arXiv preprint arXiv:1808.09131.
A. Takhirov and A. Lozovskiy, Computationally efficient modular nonlinear filter stabilization for high Reynolds number flows, Advances in Computational Mathematics, 44(1) (2018), pp. 295–325.
M. Neda, A. Takhirov, and J. Waters, Modular nonlinear filter based time relaxation scheme for high Reynolds number flows, International Journal of Numerical Analysis and Modeling, 15(4-5) (2018), pp. 699-714.
O. Isik, A. Takhirov, and H. Zheng, Second order time relaxation model for accelerating convergence to steady-state equilibrium for Navier-Stokes equations, Applied Numerical Mathematics, 119 (2017), pp. 67-78.
R. Lazarov and A. Takhirov, A note on uniform inf-sup condition for the Brinkman problem in highly heterogeneous media, Journal of Computational and Applied Mathematics, 340 (2017), pp. 537-545
A. Takhirov, Voigt regularization for the explicit time stepping of the Hall effect term, Geophysical & Astrophysical Fluid Dynamics, 1 (2016), pp. 1-23.
M. Neda, A. Takhirov, and J. Waters, Time relaxation algorithm for flow ensembles, Numerical Methods for Partial Differential Equations, 32 (2016), pp. 757-777.
W. Layton, M. Sussman and A. Takhirov, Instability of Crank-Nicolson Leap-Frog for nonautonomous systems, International Journal of Numerical Analysis and Modeling B, 5(3), pp. 289-298.
A. Takhirov, Exact solutions of stochastic Navier-Stokes equations, University of Pittsburgh Technical Report TR-MATH 13-14.
W. Layton and A. Takhirov, Energy stability of a first order partitioned method for systems with general coupling, International Journal of Numerical Analysis and Modeling B, 4(3), pp. 203-214.
A. Takhirov, Stokes-Brinkman Lagrange Multiplier/Fictitious Domain Method for Flows in Pebble Bed Geometries, SIAM Journal on Numerical Analysis, 51(5), pp. 2874-2886.
W. Layton and A. Takhirov, Numerical Analysis of Wall Adapted Nonlinear Filter Models of Turbulent Flows, Contemporary Mathematics, 586, pp. 219-229.
A. Bowers, L. Rebholz, C. Trenchea and A. Takhirov, Improved Accuracy in Regularization Models of Incompressible Flow via Adaptive Nonlinear Filtering, International Journal for Numerical Methods in Fluids, 70(7), pp. 805-828.
Direction splitting scheme for spherical shell geometries, University of Pittsburgh, Applied Math Seminar, November 12.
Direction splitting scheme for spherical shell geometries, North Carolina A&T State University, Department of Math Colloquim, March 1.
Computationally efficient adaptive nonlinear filtering scheme for high Reynolds number flows, AMS Spring Eastern Section, College of Charleston, March 11.
Computationally efficient adaptive nonlinear filtering scheme for high Reynolds number flows, Finite Element Rodeo, University of Houston, March 3-4.
Voigt regularization for the explicit time stepping of the Hall effect term, AMS Fall Western Section, University of Denver, October 8-9.
Uniform inf-sup condition for the Brinkman problem in highly heterogenous media, Numerical Analysis and Predictability of Fluid Flow, University of Pittsburgh, May 3-4.
Voigt regularization for the explicit time stepping of the Hall effect term, AMS Spring Western Section Meeting, UNLV, April 18.
Numerical Analysis of the Flows in Pebble Bed Geometries, Mathematics and Computer Science Division Seminar, Argonne National Laboratory, January 22.
Numerical Analysis of the Flows in Pebble Bed Geometries, Computer Science and Mathematics Division Seminar, Oak Ridge National Laboratory, January 16.
Ensemble calculations for time relaxation fluid flow models, PostDoc seminar, Department of Mathematics, Texas A&M University, October 21.
Improved Accuracy in Regularization Models of Incompressible Flow via Adaptive Nonlinear Filtering, Numerical Analysis seminar, Department of Mathematics, Texas A&M University, October 1.
Stokes-Brinkman Lagrange Multiplier/Fictitious Domain Method for Flows in Pebble Bed Geometries, 5th Clemson University SIAM Chapter Conference, Clemson University.
Stokes-Brinkman Lagrange Multiplier/Fictitious Domain Method for Flows in Pebble Bed Geometries, Mid-Atlantic Numerical Analysis Day, Temple University.
Stokes-Brinkman Lagrange Multiplier/Fictitious Domain Method for Flows in Pebble Bed Geometries, Finite Element Circus, University of Pittsburgh.
Stokes-Brinkman Lagrange Multiplier/Fictitious Domain Method for Flows in Pebble Bed Geometries, 2012 SIAM Annual Meeting, Minneapolis.
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems, Math 2602 - Mixed Finite Element Methods, University of Pittsburgh, April.
Improved Accuracy in Regularization Models of Incompressible Flow via Adaptive Nonlinear Filtering , Mid-Atlantic Numerical Analysis Day 2011, Temple University.
Ocean-atmospheric flows, astrophysical flows and geodynamo occur in spherical shells. However, the standard spherical transformation suffers from grid convergence near the z-axis and results in numerical inefficiency. My current work uses composite latitude-longitude Yin-Yang spherical transformation. Via direction splitting approach, solving the original 3D problem is replaced with a sequence of 1D problems, which in turn are discretized in space using centered finite differences. All resulting systems are tridiagonal, which are directly solved using the domain decomposition induced Schur complement technique. The entire procedure is implemented in parallel using MPI.
Computing an ensemble of turbulent flow equations with perturbed/noisy data is a common procedure in many engineering and geophysical applications to quantify uncertainty and make predictions. In a recent work, I developed an ensemble calculation scheme for incompressible Navier-Stokes equations with provable energy stability for domains with outflow boundaries, a common scenario when a large or physically-unbounded domain is truncated. Given ensemble of perturbed initial conditions, forcing terms and viscosities, the scheme allows for the calculation of the ensemble solutions by inverting multiple linear systems with a single matrix. Stability and convergence hold under a mild CFL condition involving the velocity fluctuations. The numerical tests for moderate Reynolds number flows gave results that are very close to those of independent simulations.
Turbulence is the natural state of many flows. Brute-force numerical simulation of many turbulent flows is still a challenge, and legacy codes play a central role in practical simulations of turbulent flows. As our understanding of the field grows and more accurate methods are developed, one challenge is incorporating these new advances into the old computing platforms. My current work is development of nonlinear adaptive filtering schemes that allow incorporating modern turbulence models into the existing turbulence and multiphysics codes in a modular fashion.
It is believed that the magnetic field of the planets and stars is generated by the fluid dynamo action, a process through which the flow of electrically conducting fluid maintains the magnetic field. Dynamo has also been recently obtained in laboratory experiments in cylindrical containers in Riga, Karlsruhe, and Cadarache. However, the Madison Dynamo Experiment failed to produce the expected dynamo in a spherical region. My current work is on determination of physical parameters that would produce a dynamo in a sphere using Hybrid Fourier/Finite Element approach.
Differential Equations (Most recent student evaluations)
Engineering Math I, II
Intro to Theoretical Math
Calculus I, II, III