Around 1967, Kraichnan, Leith, and Batchelor (KLB) independently proposed the dual cascade theory, which is thought to describe turbulence in unbounded two-dimensional fluids. In a bounded domain, however, the upscale energy cascade they discussed will be halted at the lowest wavenumber (corresponding to the domain size). An upper bound on the ratio of the total enstrophy to total energy derived by Tran and Shepherd [Physica D, 2002] establishes that the energy must be dissipated at scales larger than the forcing scale. This result is based on the assumption that the square root of the ratio of mean enstrophy to mean energy injection is spectrally localized to the forcing region. We investigate the conjecture that turbulence driven by a spectrally localized temporally white-noise random forcing satisfies this assumption. We also provide numerical evidence that energetic reflections at the lower spectral boundary may eventually lead to a large-scale k-3 energy spectrum, in agreement with the large-scale k-3 spectra observed in the atmosphere by Lilly and Peterson [Tellus 35A, 379 (1983)]. A spectral constraint derived by Tran and Bowman [Physica D, 2003] establishes that the two inertial-range exponents must sum to -8. A large-scale k-3 spectrum resulting from reflections at the lower spectral boundary would then explain the small-scale k-5 spectrum frequently observed in numerical simulations of the enstrophy range. We propose that combined supergrid and subgrid models based on Kolmogorov's hypothesis of self-similar energy (or enstrophy) transfer could be used to mimic the behaviour of an unbounded fluid in a doubly periodic domain, thereby allowing one to address the validity of the classical KLB theory for unbounded fluids.
The lecture begins with a short review of physical background of composite materials and porous media like periodicity or randomness (from static and kinetic mechanical problems of composite materials, heat and mass transfer in porous media, wave propagation in heterogeous media, etc.), and then introduces several recent mathematical results on the asymptotic homogenization methods. In particular, I would like to advance our methods and numerical results for the above physical problems. Finally, some open problems are also presented in corresponding sections.
New efficient algorithms are formulated and analyzed for the solution of a class of linear partial integro-differential equations of parabolic type in the unit square. In these methods, orthogonal spline collocation (OSC) with C1 piecewise polynomials of degree ³ 3 is used for the spatial discretization. For the time stepping, alternating direction implicit (ADI) methods based on the backward Euler method, the Crank Nicolson method and the second order BDF scheme are considered. Such methods reduce the multidimensional problem to sets of independent one-dimensional problems in which the OSC matrices are easily determined since, unlike the finite element Galerkin case, no integrals must be evaluated or approximated. The methods are shown to be of first or second order accuracy in time and of optimal order accuracy in the L2, H1 and H2 norms in space. From the analysis, a new optimal order H2 estimate is obtained for an ADI OSC Crank Nicolson method for the heat equation in two space variables. ADI OSC methods are also examined for the solution of a class of evolution equations with a positive type memory term, and an optimal order L2 estimate is derived for each method.
This is joint work with Amiya Pani and Bernard Bialecki.
There are numerous challenges faced by scientists when developing algorithm libraries for scientific applications. Issues of functionality, stability, performance, and flexibility depending on the specific application areas are at the focus of researchers working in both theoretical and applied areas. This talk discusses some of the approaches to development and implementation of two- and three-dimensional data structures for applications in molecular biology, mechanical engineering and GIS. The talk concentrates on issues of algorithm efficiency, precision of the result and numerical stability. Challenges in the development of the ECL (Exact Computational Library) for performing exact computations in the fixed precision floating-point arithmetic are discussed. The ECLibrary is based on the interval point arithmetic, iterative approximation methods, reduction technique and algorithms for performing complex transformations on floating-point numbers. The ESAE algorithm (Exact Sign of Algebraic Expression) will be described. Some implementation issues will be also discussed.
The use of standard numerical discretization techniques, such as explicit RK methods, to integrate non-linear differential equations often leads to scheme-dependent instabilities and/or convergence to spurious solutions when certain step-sizes or parameter values are used in their simulations. This paper presents some nonstandard finite-difference methods that are, in general, free of the aforementioned drawbacks. These schemes are designed in such a way that they preserve the important features/properties of the continous model they approximate.
The convection diffusion equations, which describe many realistic procedures in many problems of science and technology; eg., fluid mechanics, heat and mass transfer, groundwater modelling, petroleum reservoir simulation and environmental protection, are very important and difficult in numerical simulation. The standard finite difference methods or finite element methods will introduce severe nonphysical oscillations into the numerical solutions since the corresponding discrete schemes are unstable for the problems. Because of satisfying both the stability and the conservation of mass, the methods of finite-volume-type with upwinding techniques have obtained high successes in the numerical simulation of the convection diffusion problems. However, the standard upwinding technique treating convection terms usually derives lower-order accuracy schemes for the problems. In this talk, we will present the nonstandard high-order upwinding finite covolume methods for the convection diffusion problems. The conservation law of mass and the unconditional stability are analyzed, the high-order error estimates are obtained for the methods. Numerical experiments are given to demonstrate the performance of the schemes.
We will discuss an immersed finite element method for axial symmetric three dimensional nonlinear interface problems. The basis functions in this method are piece-wise linear polynomials satisfying the jump conditions approximately (or even exactly in many situations). In addition, the mesh in this method does not have to be aligned with the interface because the interface is allowed to pass through the elements. Therefore, structured Cartesian meshes can be used in this method to facilitate efficient numerical solutions. We will show that this method has the usual second and first order convergence rates in L2 and H1 norms, respectively. Numerical examples for a nonlinear interface problem arising from ion optics modelling in composite structures will be provided to illustrate features of this method.
In this talk we will survey on recent results on nonconforming finite elements on quadrilaterals (http://www.nasc.snu.ac.kr/). These elements, originally developed for elliptic problems, are applied to solving Stokes, elasticity, Helmholtz, and Maxwell's equations. Error estimates and selected numerical results will be shown.
Although interest in the biomechanics of the brain goes back over centuries, mathematical models of hydrocephalus and other brain abnormalities are still in their infancy and a much more recent phenomenon. This is rather surprising, since hydrocephalus is still an endemic condition in the pediatric population. Treatment has dramatically improved over the last 30 years thanks to the introduction of CSF-shunts. This too, however, is not without problems and the shunt failure rate at 2 years post shunt insertion is 50common causes of shunt failure is due to shunt obstruction. Common sense suggests that the optimal shunt location would be in the ventricular region that remains largest after ventricular decompression drainage. In this talk, we will report on some recent progress towards the solution of this problem.
An adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two-point boundary value problems. We combine a multiquadric integral formulation with a previously employed coordinate stretching technique. A new error indicator function accurately captures the regions of the interval with insufficient resolution. This indicator is used to adaptively add data centres and collocation points. Our method resolves extremely thin layers accurately with fairly few basis functions. We demonstrate the effectiveness and the robustness of our new method on a number of examples.
Joint work with Leevan Ling, SFU.
We consider spatially discrete FitzHugh-Nagumo equations as a model for ionic conductances that generate the action potential of nerve fibers in motor nerves of vertebrates. Existence results for front and pulse solutions are discussed. Numerical techniques are considered to approximate solutions to mixed type functional differential equations obtained when considering traveling fronts and pulses of these equations. Extensions to coupled systems of spatially discrete FitzHugh-Nagumo equations corresponding to bundles of nerve fibers will be discussed.
A new gradient recovery technique (PPR) is introduced to construct a posteriori error estimator. The recovery operator is polynomial preserving and insensitive to mesh distortion. Some theoretical results are provided regarding the recovery operator. In addition, some numerical results are provided in comparison with the Zienkiewicz-Zhu error estimator based on the superconvergent patch recovery (SPR).