Divakar Viswanath

Department of Mathematics
University of Michigan
530 Church Street
Ann Arbor, MI 48103
Much of my research is centered around numerical algorithms. An ongoing effort is to derive solvers of the Navier-Stokes equation at high Reynolds numbers. The incompressible Navier-Stokes equation captures a great variety of phenomena and develops turbulent solutions at even low Reynolds numbers. Computing turbulent solutions at high Reynolds numbers requires grids with more than a billion points. Such large simulations are now possible on even a single computer or two. It is a common belief that computer simulations become more accurate as grids become finer. This belief is true most of the time, but when the grids become extremely fine, rounding errors and not discretization errors begin to dominate. Rounding errors increase as grids become finer. I have been deriving solvers that contain the effect of rounding errors and allow accurate simulations with finer and finer grids.
A part of my research has dealt with the interface to dynamical systems theory. Here I have developed methods that exhibit the fractal structure of the iconic Lorenz attractor in detail. Another contribution is a method for finding periodic orbits and other solutions of partial differential equations that are applicable to transitional turbulence.
In the past, I have worked on problems in probability theory. One result in this area deals with the rate of growth of random Fibonacci sequences. I have also worked on problems related to mixing times of Markov chains. A significant result related to card shuffling was obtained by Mark Conger who discovered a way to deal cards in bridge that appears to be a substantial improvement of the usual cyclic deal.
I have a general interest in fundamental problems of numerical analysis, which includes interpolation and approximation of functions and their derivatives. Most recently, I have become interested in connections to the Vapnik-Smale approach to statistical learning algorithms.
I hold a bachelor’s degree from the Indian Institute of Technology at Bombay and a PhD degree from Cornell University, both in Computer Science. I am Professor of Mathematics at the University of Michigan.
[26] R. Navarrete and D. Viswanath, Accuracy and stability of inversion of power series, submitted. Arxiv.
[25] D. Viswanath, Navier-Stokes solver using Green’s functions II: spectral integration of channel flow and plane Couette flow, submitted. Arxiv.
[24] D. Viswanath, Spectral integration of linear boundary value problems, submitted. Arxiv.
[23] D. Viswanath and I. Tobasco, Navier–Stokes solver using Green’s functions I: Channel flow and plane Couette flow, Journal of Computational Physics, vol. 251 (2013), p. 414–431. Arxiv.
[22] B. Sadiq and D. Viswanath, Barycentric Hermite interpolation, SIAM Journal on Scientific Computing, vol. 35 (2013), p. A1254-A1270. Arxiv.
[21] D. Viswanath, K. Serkh and X. Liang, Metric entropy and the optimal prediction of chaotic signals, SIAM Journal on Applied Dynamical Systems, vol. 12 (2013), p. 1085-1113. Arxiv.
[20] B. Sadiq and D. Viswanath, Finite difference weights, spectral differentiation, and superconvergence, Mathematics of Computation, vol. 83 (2014), p. 2403–2427. Arxiv.
[19] Y. Charles Li and D. Viswanath, Exact and asymptotic conditions on traveling wave solutions of the Navier-Stokes Equations, Physics of Fluids, vol. 21 (2009), 101703. Link: PDF.
[18] D. Viswanath and S. Şahutoğlu, Complex singularities and the Lorenz attractor, SIAM Review, vol. 52 (2010), p. 294-314. Arxiv.
[17] D. Viswanath and P. Cvitanović, Stable manifolds and the transition to turbulence in pipe flow, J. Fluid Mech., vol. 627 (2009), p. 215-233. Arxiv.
[16] J. Halcrow, J.F. Gibson, P. Cvitanović and D. Viswanath, Heteroclinic connections in plane Couette flow, J. Fluid Mech., vol. 611 (2009), p. 365-376. Link: PDF.
[15] D. Viswanath, The critical layer in pipe flow at high Reynolds number, Phil. Trans. Roy. Soc., vol. 367 (2009), p. 561-576. Arxiv.
[14] D. Viswanath, The dynamics of transition to turbulence in plane Couette flow, Mathematics and computation: a contemporary view. The Abel Symosium 2006. Springer-Verlag. Berlin. 2008. Arxiv.
[13] D. Viswanath, Recurrent motions within plane Couette turbulence, J. Fluid Mech., 580 (2007), 339-358. Arxiv.
[12] M. Conger and D. Viswanath, Shuffling cards for blackjack, bridge, and other card games. Arxiv.
[11] D. Viswanath, Stable manifolds and homoclinic points near resonances in the restricted three-body problem, Celestial Mechanics and Dynamical Astronomy, 94 (2006), 213-235. Arxiv.
[10] M. Conger and D. Viswanath, Normal approximations for descents and inversions of permutations of multisets, J. Theor. Probab., 20 (2007), 309-325. Arxiv.
[9] D. Viswanath, Linear stability analysis of resonant periodic motions in the restricted three-body problem, . Dynamics and Differential Equations, 17 (2005), 271-292. Arxiv.
[8] M. Conger and D. Viswanath, Riffle shuffles of decks with repeated cards, Annals of Probability, 34 (2006), 804-819. Arxiv.
[7] D. Viswanath, The fractal property of the Lorenz attractor, Physica D, 190 (2004), 115-128. Link: PDF
[6] D. Viswanath, Symbolic dynamics and periodic orbits of the Lorenz attractor, Nonlinearity, 16 (2003), 1035-1056. Link: PDF.
[5] D. Viswanath, How many timesteps for a cycle? Analysis of the Wisdom-Holman algorithm?, BIT, 42 (2002), 194-205.
[4] D. Viswanath, The Lindstedt-Poincaré technique as an algorithm for computing periodic orbits, IAM Review, 43 (2001), 479-495. Link: PDF
[3] D. Viswanath, Global errors of numerical ODE solvers and Lyapunov’s theory of stability, IMA J. Numer. Anal., 21 (2001), 387-406. Link: PDF
[2] D. Viswanath, Random Fibonacci sequences and the number 1.13198824..., Math. Comp., 69 (2000), 1131-1155. Link: PDF
[1] D. Viswanath and L.N. Trefethen, Condition numbers of random triangular matrices, SIAM J. Matrix. Anal. Appl., 19 (1998), 564-581. Link: PDF.