Department of Mathematics

Applied and Interdisciplinary Mathematics Seminar

Fall 2000 (Winter 2001)

The seminar meets on Friday, 3pm , in 3096 East Hall unless otherwise noted.



Questions? Comments? Please email kurganov@math.lsa.umich.edu
or
petrova@math.lsa.umich.edu .



  • Friday, September 8
    Hans Johnston, University of Michigan
    "APPS: A Novel Scheme for the Incompressible Navier-Stokes Equations based on an Accurate Pressure Poisson Solver."

    We present a second order numerical method for the time dependent incompressible Navier-Stokes in the velocity-pressure formulation implemented on non-staggered grids. The key to the method lies in a consistent and accurate approximation of the Neumann boundary condition for the pressure Poisson equation. With this numerical boundary condition we are able to treat the pressure as a dynamic variable in the time evolution of the flow. The resulting scheme is simple to implement and highly efficient. Normal mode analysis and numerical examples will be presented to demonstate the accuracy and convergence properties of the scheme. Extension of the overall method to both variable density and 3D flows will also be discussed.

  • Friday, September 15
    Paul Newton, University of Southern California
    "Vortex Motion and Global Streamline Patterns on a Sphere"

    The talk will describe the evolution of vorticity on a sphere and the corresponding velocity fields and streamline patterns that are produced. Our analysis is motivated by the paradox that despite the fact that large scale geophysical vortex structures such as atmospheric cyclones or oceanographic eddies usually generate turbulent flowfields, many typical averaged streamline patterns produced by these structures are relatively simple. Because of the large ratio of horizontal to vertical length scales inherent in these structures, the two-dimensional Hamiltonian formulation of the Euler equations can be used. Because the structures typically survive for long times and are capable of transporting passive scalars such as heat, environmental pollutants, or biota over large distances, the spherical geometry of the Earth's surface becomes important.
    The first part of the talk will introduce the basic equations of motion for point vortices on a sphere based on the two dimensional Euler equations in spherical geometry, emphasizing the Hamiltonian structure of the problem and the special role played by triad interactions. I will briefly describe relevant aspects of our recent solution to the integrable three vortex problem, including equilibrium configurations and their nonlinear stability, and non-equilibrium configurations (periodic and quasi-periodic solutions and finite time collapsing states).
    The main part of the talk will focus on our recent categorization of all generic integrable streamline patterns produced by the vortices. In particular, we have shown that there are exactly 12 topologically distinct instantaneous streamline patterns produced by three vortices, which then can be continuously deformed to form an additional 23 distinct but homotopically equivalent structures. Using these 35 patterns as templates, we show how spherical streamline plots from atmospheric weather data can be decomposed and their patterns identified. We finish the talk with speculation on why such relatively simple patterns might be generic in real atmospheric flows, based on a statistical mechanical argument.
    Future challenges include understanding the evolution of streamline patterns on the sphere and the role that the instantaneous structures play in the finite time mixing and transport of passive particles, as well as the addition of other effects such as the added complication of coastlines and topography for oceanographic flows.

  • Friday, September 29
    Nick Ercolani, University of Arizona
    "Landau Theory for Irrotational Vector Fields"

    The singular perturbation of the potential energy $\int(1-u^2)^2$ by $\epsilon^2 |\nabla u|^2$ is a classical model for phase transitions. The extension of this problem from scalar fields $u$ to gradient vector fields has until recently resisted analysis. In this talk we will review some of the physical motivations for this latter problem coming, primarily, from the modelling of defects in pattern formation. We will also describe the derivation of the associated variational models and aspects of their singular limits.

  • Friday, October 6
    Alina Chertock, UC Berkeley
    "Self-Similar Intermediate Asymptotics for a Degenerate Parabolic Filtration-Absorption Equation"

    The equation $$ \u_t=uu_{xx}-(c-1)(u_x)^2 $$ is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water absorbing fissurized porous rock, therefore we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.

  • Friday, October 13
    Alexander Kurganov, University of Michigan
    "Local Error Analysis and Adaptive Semi-Discrete Central Schemes for Hyperbolic Conservation Laws"

    We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of computing the truncation error in the local Lip'-seminorm through the use of localized test-functions. In the convex scalar case, the derived local truncation error estimates can be converted into pointwise ones (following the Lip' convergence theory developed by Tadmor et al.)
    Our numerical experiments show remarkably similar behavior of the local Lip' truncation error and the actual error even in the case of systems of conservation laws. The difference of several orders of magnitude in the local truncation errors between smooth and nonsmooth regions provides an effective tool for identifying nonsmooth parts of the solution. This may be utilized in scheme- and mesh-adaption algorithms, and is the key idea in developing adaptive semidiscrete schemes.
    The main building block of the proposed adaptive methods is the semidiscrete, second- and third-order, Godunov-type, Riemann-solver-free central schemes. The use the local Lip' truncation error as an error indicator allows us to additionally increase the efficiency of the central schemes by not applying any costly nonlinear limiters in the smooth regions. This leads to a scheme-adaptive central method -- different piecewise polynomial reconstructions (and thus, different central schemes) are used in smooth and nonsmooth domains.
    The same error indicator is also utilized for incorporating the mesh-adaption technique into the constructed method. This further reduces the overall cost of the proposed adaptive method, and makes our central schemes the most powerful `black-box-solvers' for hyperbolic systems of conservation laws.

  • Friday, October 20
    Yuji Kodama, Ohio State University
    "Toda Lattices, Bruhat Decomposition and Topology of Iso-spectral Real Manifolds"

    I start to present a result of the total number of zeros for the Hankel determinants generated by an exponential polynomial. Then I will explain how this result connects to the Toda lattice equations with indefinite metric, the Bruhat decomposition of the algebraic linear group and topology of the iso-spectral real manifold of tri-diagonal matrices with fixed eigenvalues. The talk is based on collaborations with J. Ye and L. Casian.

  • Friday, October 27
    Jonathan Mattingly, Stanford University
    "Ergodicity in the 2D Stochastically Forced Navier Stokes Equation"

    One basic assumption in the theory of turbulence is that under large scale forcing, energy is transferred throguh nonlinearity to the small scales and the system establishes a unique statistical steady state. In numerical simulations, one typically forces very few low modes when studying the direct cascade process. Statistical properties of the turbulent flow are measured or calculated through time averaging rather than ensemble averaging.
    One main purpose of this talk is to rigorously establish the validity of this basic assumption. We find it convenient to study this problem in a stochastic setting.
    The ergodic theory of stochastic PDEs is delicate and poorly understood (when compared with stochastic ODEs). I will describe a technique which gives a number of new results for the 2D Stochastically Forced Navier Stokes Equation. It mixes a dynamic and a statistical understanding of the equation, building on ideas from infinite dimensional dynamical systems, statistical mechanics, and Markov chain theory. One could alternatively describe the approach as coupling with infinite memory or as the lack of phase transitions when correlations decay exponentially.
    Along the way I will explain how there are different mechanisms for convergence in different regimes and how this is the key to making progress. I will use a number of simple examples to illustrate these points.

  • Friday, November 3
    N. Wang, University of Minnesota
    "Ambient fields in systems with many particles"

    In many cases of systems with many particles, it is known that each particle's behavior depends on some "ambient quantities" it detects. Traditional techniques that are used (for example, space averaging) have not been able to identify what exactly these ambient quantities are. In this talk, we will introduce a new method with which ambient quantities can be rigorously defined. We will use bubbly flow, effective thermal conductivity of heterogeneous media as examples to show how the method can yield some new results. It also seems that the new method could have some applications in certain aspects of classical physics.

  • Friday, November 10
    Daniel Szyld, Temple University
    "Asynchronous Parallel Iterative Methods for Linear and Nonlinear Problems"

    Asynchronous iterations arise naturally on parallel computers if one wants to minimize idle times and load balancing is either impractical or not advisable (e.g. when one subdomain of a differential equation needs a much finer resolution than the others).
    In these methods, local problems are solved in each processor with the information available at the time, i.e., without waiting for the other processors to complete their tasks. In other words, some of the information may be "old". Nevertheless, since there is no idle time, convergence is attained in less time than if a synchronization bottleneck has to be crossed.
    In this talk, certain models of asynchronous iterations are reviewed using a common theoretical framework. The corresponding convergence theory for linear systems, nonlinear systems, and initial value problems is described.
    Instances where computational experiments have been reported in the literature in various fileds of applications, and which illustrate the potential for these methods, are presented.

  • Friday, November 17
    Derek Abbott, Adelaide University
    "Parrondo's Paradox: When Losing Strategies Cooperate to Win"

    It is well known that chess pieces can be sacrificed in order to win the overall game. In population genetics, the fitness of an animal species sometimes declines first before evolving to a higher level of survival fitness. Engineers know that unstable control systems paradoxically stabilize if combined in the right way. However, can losing gambling games be constructed so they become winning, when they are played one after each other? The answer is yes. Game A and game B are constructed to be losing if played individually, but if A and B are played in any random sequence we get a winning expectation! This strikingly counterintuitive result is called Parrondo's Paradox. We will take a look at some simple examples and show that this effect is in fact a discrete-time manifestation of a Brownian ratchet. Game theory is strongly linked to economics, social networks/dynamics and evolutionary biology. Future development of Parrondian strategies that can help to model cases where declining birth-death processes combine in a beneficial way are thus of great interest. In this talk, there will be something of interest to virtually all scientific disciplines: physicists, biologists, engineers, mathematicians and economists will all be catered for.

  • Friday, December 1
    Jan Vecer, University of Michigan
    "Trading with Safety Net"

    In this talk, we study the case when the trader wants to be insured for the case of the loss resulting from his trading strategy. This is a typical situation which financial institutions (banks, investment funds, pension funds or hedging funds) have to face: they are implementing some trading strategy which include active trading in the stock and the money market. Since these institutions are in some extent responsible for the outcome of their trading, it is desirable that they would have only limited liability when the loss occurs. They can buy an option contract which would let them to keep the profits, but they would be forgiven the loss. Using the probability techniques combined with the techniques of the stochastic optimal control (Hamilton-Jacobi-Bellman equations), we can compute price of this type of product. Moreover, we can determine what is the optimal trading strategy for the holder of the contract and the hedging strategy for the seller of the contract. In many cases, the price of this type of insurance is surprisingly cheap.

  • Friday, December 8
    Robert Krasny, University of Michigan
    "A Particle Method and Adaptive Treecode for Vortex Sheet Motion in 3-D Flow"

    A particle method is presented for computing vortex sheet motion in three-dimensional flow. The particles representing the sheet are advected using the Rosenhead-Moore form of regularized Biot-Savart kernel and new particles are inserted during the simulation to maintain resolution as the sheet rolls up. The particle velocities are computed using a treecode based on Taylor approximation in Cartesian coordinates. The necessary coefficients are computed using a recurrence relation and several adaptive techniques are employed to gain efficiency including nonuniform rectangular cells, variable order approximation, and a run-time choice between approximation and direct summation. Tests show that the treecode is significantly faster than direct summation for systems having a large number of particles. The algorithm is applied to simulate flows in which a vortex sheet rolls up to form a vortex ring. Two examples are presented: (1) azimuthal waves on a vortex ring, and (2) merger of two vortex rings.
    (This is joint work with Keith Lindsay, NCAR)