Contact Information
Divakar Viswanath
530 Church Street
Department of Mathematics
University of Michigan
Ann Arbor, MI 48109, U.S.A.
Office: East Hall 3856
Phone: (734) 936-9995
Email: divakar[AT]umich.edu
Classes
Math 572: Introduction to Scientific Computing II
Publications:
# Title Summary Reference Dwnld
[21] Barycentric Hermite interpolation Numerical stability of barycentric Hermite interpolation even when derivatives are of high order. New method to update barycentric coefficients in linear time. Sadiq and DV, submitted. pdf
[20] Metric Entropy and the Optimal Prediction of Chaotic Signals We show that all current methods for predicting chaotic signals are necessarily suboptimal. We make a firm and significant connection between the problem of predicting chaotic signals and a major result in modern information theory. We investigate ideas that point the way to the derivation of an optimal general purpose predictor for chaotic signals. DV, Kirill Serkh, and Xuan Liang, submitted. pdf
[19] Finite Difference Weights Using The Modified Lagrange Interpolant A new method for computing finite difference weights. Basic result saying that the order of accuracy of a finite difference formula cannot be boosted by more than one beyond its typical order of accuracy. Algebraic condition for boost in order of accuracy. Burhan Sadiq and DV, submitted.

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[18] Exact and Asymptotic Conditions on Traveling Wave Solutions of the Navier-Stokes Equations Some conditions derived are exact and must hold for any traveling wave solution or periodic solution irrespective of the Reynolds number (Re). Other conditions are asymptotic in the limit of large Re. The latter are verified numerically. Y. Charles Li and DV, Physics of Fluids, vol. 21 (2009), 101703. pdf
[17] Complex singularities and the Lorenz attractor. We suggest that the mathematical analysis of the Lorenz system could be a problem in analytic function theory. A complete and consistent formal development of singular solutions of the Lorenz system is given. Convergence is proved using a technique that is simpler, more transparent and more powerful than that of Hille. DV and S. Şahutoğlu, SIAM Review, vol. 52 (2010), p. 294-314.. pdf
[16] Stable manifolds and the transition to turbulence in pipe flow A numerical method is devised for disturbing the laminar motion in such a way that it hits a given traveling wave while transitioning to turbulence. The connection of exact solutions of the Navier-Stokes equation to experiment is discussed. DV and P. Cvitanović, J. Fluid Mech., vol. 627 (2009), p. 215-233. pdf
[15] Heteroclinic connections in plane Couette flow A numerical method for locating heteroclinic connections, illustrations of the change in streak patterns along some connections, and phase space portraits. J. Halcrow, J.F. Gibson, P. Cvitanović and DV, J. Fluid Mech., vol. 611 (2009), p. 365-376. pdf
[14] The critical layer in pipe flow at high Reynolds number Computations up to Re=75000. Details of numerical method for locating traveling waves and relative periodic orbits. DV, Phil. Trans. Roy. Soc., vol. 367 (2009), p. 561-576. pdf
[13] The dynamics of transition to turbulence in plane Couette flow Possible role of lower branch solutions in transition to turbulence. [in] Mathematics and computation: a contemporary view. The Abel Symosium 2006. Springer-Verlag. Berlin. 2008. pdf
[12] Recurrent motions within plane Couette turbulence. Relative periodic orbits that correspond to coherent motions in the near-wall region of turbulent boundary layers. Importance of spanwise drift. Outline of numerical method for finding relative periodic or traveling wave or equilibrium solutions of the Navier-Stokes equation. DV, J. Fluid Mech., 580 (2007), 339-358. pdf
[11] Shuffling cards for blackjack, bridge, and other card games The number of shuffles to mix a deck depends upon the way the cards are dealt (in bridge). Determining the total variation distance is NP complete. Excellent estimates are computed and partially justified in a variety of situations. Under revision pdf
[10] Stable manifolds and homoclinic points near resonances in the restricted three-body problem Proof of existence of homoclinic points near resonances DV, Celestial Mechanics and Dynamical Astronomy, 94 (2006), 213-235. pdf
[9] Normal approximations for descents and inversions of permutations of multisets In a long list of numbers, each instance where a larger number precedes (immediately) a smaller number is an inversion (descent). Central limit theorems for descents and inversions. Application to DNA sequences. M. Conger and DV, J. Theor. Probab., 20 (2007), 309-325. pdf
[8] Linear stability analysis of resonant periodic motions in the restricted three-body problem Resonant periodic motions are alternately stable and unstable. DV, J. Dynamics and Differential Equations, 17 (2005), 271-292. pdf
[7] Riffle shuffles of decks with repeated cards Descent polynomials for decks with repeated cards. Combinatorics and algorithms for descent polynomials. M. Conger and DV, Annals of Probability, 34 (2006), 804-819. pdf
[6] The fractal property of the Lorenz attractor Detailed computations of the fractal structure of the Lorenz attractor. Numerical method for extended precision computation of periodic orbits in which most of the arithmetic uses double precision. DV, Physica D, 190 (2004), 115-128. pdf
[5] Symbolic dynamics and periodic orbits of the Lorenz attractor Computation of orbits with special symbol sequences. Numerical demonstration of the density of periodic orbits in a strange attractor. First computation of the fractal structure of the Lorenz attractor deduced by Lorenz in 1963. Appendix describes an essential extension to the numerical method of [3]. DV, Nonlinearity, 16 (2003), 1035-1056. pdf
[4] How many timesteps for a cycle? Analysis of the Wisdom-Holman algorithm? DV, BIT, 42 (2002), 194-205.
[3] The Lindstedt-Poincaré technique as an algorithm for computing periodic orbits Computation of periodic orbits reduces to a two point boundary value problem (BVP). The method described has two novelties: the nonlinear BVP is linearized before it is discretized and the Lindstedt-Poincaré device is used to find the period. The method is especially accurate (as illustrated further in [5]). DV, SIAM Review, 43 (2001), 479-495. pdf
[2] Global errors of numerical ODE solvers and Lyapunov's theory of stability Linear upper bounds for global error of trajectories that converge to stable structures such as fixed points, periodic orbits, and invariant torii. DV, IMA J. Numer. Anal., 21 (2001), 387-406. pdf
[1] Random Fibonacci sequences and the number 1.13198824... Asymptotic growth rate of random Fibonacci sequences. DV, Math. Comp., 69 (2000), 1131-1155. pdf
[0] Condition numbers of random triangular matrices Random triangular matrices are exponentially ill-conditioned. Asymptotically exact formulas for the condition number are determined in several situations. DV and L.N. Trefethen, SIAM J. Matrix. Anal. Appl., 19 (1998), 564-581. pdf