Showing papers in "Communications in Mathematical Sciences in 2003"

Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis

Abstract: We present and discuss a framework for computer-aided multiscale analysis, which enables models at a fine (microscopic/stochastic) level of description to perform modeling tasks at a coarse (macroscopic, systems) level. These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains. Traditional modeling approaches first involve the derivation of macroscopic evolution equations (balances closed through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem. Our equation-free (EF) approach, introduced in (1), when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales,"coarse" bifurcation analysis, optimization, and control) directly. In effect, the procedure constitutes a system identification based, "closure-on-demand" computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We will briefly survey the application of these "numerical enabling technology" ideas through examples including the computation of coarsely self-similar solutions, and discuss various features, limitations and potential extensions of the approach.

The Heterognous Multiscale Methods

Abstract: The heterogenous multiscale method (HMM) is presented as a general methodology for the efficient numerical computation of problems with multiscales and multiphysics on multigrids. Both variational and dynamic problems are considered. The method relies on an efficent coupling between the macroscopic and microscopic models. In cases when the macroscopic model is not explicity available or invalid, the microscopic solver is used to supply the necessary data for the microscopic solver. Besides unifying several existing multiscale methods such as the ab initio molecular dynamics [13], quasicontinuum methods [73,69,68] and projective methods for systems with multiscales [34,35], HMM also provides a methodology for designing new methods for a large variety of multiscale problems. A framework is presented for the analysis of the stability and accuracy of HMM. Applications to problems such as homogenization, molecular dynamics, kinetic models and interfacial dynamics are discussed.

Semi-implicit spectral deferred correction methods for ordinary differential equations

Abstract: A semi-implicit formulation of the method of spectral deferred corrections (SISDC) for ordinary differential equations with both stiff and non-stiff terms is presented. Several modifications and variations to the original spectral deferred corrections method by Dutt, Greengard, and Rokhlin concerning the choice of integration points and the form of the correction iteration are presented. The stability and accuracy of the resulting ODE methods are explored analytically and numerically. The SISDC methods are intended to be combined with the method of lines approach to yield a flexible framework for creating higher-order semi-implicit methods for partial differential equations. A discussion and numerical examples of the SISDC method applied to advection-diffusion type equations are included. The results suggest that higher-order SISDC methods are more efficient than semi-implicit Runge-Kutta methods for moderately stiff problems in terms of accuracy per function evaluation.

Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability

Abstract: This paper is concerned with transparent boundary conditions (TBCs) for the time-dependent Schrodinger equation in one and two dimensions. Discrete TBCs are introduced in the numerical simulations of whole space problems in order to reduce the computational domain to a finite region. Since the discrete TBC for the Schrodinger equation includes a convolution w.r.t. time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate TBCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove stability of the resulting initialboundary value scheme, give error estimates for the considered approximation of the boundary condition, and illustrate the efficiency of the proposed method on several examples.

Numerical techniques for multi-scale dynamical systems with stochastic effects ⁄

Abstract: Numerical schemes are presented for dynamical systems with multiple time-scales. Two classes of mehtods are discussed, depending on the time interval which the evolution of the slow variables in the system is sought. On rather short time intervals, the slow variables satisfy ordinary differential equations. On longer time intervals, however, fluctuations become important, and stochastic differential equations are obtained. In both cases, the numerical methods compute the evolution of the slow variables without having to derive explicitly the effective equations beforehand; rather, the coefficients entering these equations are obtained on the fly using simulations of appropriate auxiliary systems.

REVIEW ARTICLE: Level Set Methods and Their Applications in Image Science

Abstract: In this article, we discuss the question "What Level Set Methods can do for image science". We examine the scope of these techniques in image science, in particular in image segmentation, and introduce some relevant level set techinquies that are potnetially useful for this class of applications. We will show that image science demands multi-disciplinary knowledge and flexible but still robust methods. That is why the Level Set Method has become a thriving technique in this field.

Gauge method for viscous incompressible flows

Abstract: We present a new formulation of the incompressible Navier-Stokes equation in terms of an auxiliary field that diers from the velocity by a gauge transformation. The gauge freedom allows us to assign simple and specific boundary conditions for both the auxiliary field and the gauge field, thus eliminating the issue of pressure boundary condition in the usual primitive variable for- mulation. The resulting dynamic and kinematic equations can then be solved by standard methods for heat and Poisson equations. A normal mode analysis suggests that in contrast to the classi- cal projection method, the gauge method does not suer from the problem of numerical boundary layers. Thus the subtleties in the spatial discretization for the projection method are removed. Con- sequently, the projection step in the gauge method can be accomplished by standard Poisson solves. We demonstrate the eciency and accuracy of the gauge method by several numerical examples, including the flow past cylinder. 1. The gauge formulation In this paper, we introduce a new formulation of the incompressible Navier-Stokes equation and demonstrate that this new formulation is particularly suited for numer- ical purpose. We start with the classical formulation of the Navier-Stokes equation: ‰ u t + (u ¢ r)u + rp = 1 Re 4u

Analysis of the heterogeneous multiscale method for ordinary differential equations

Abstract: studied in averaging methods [1]. Here f and g are assumed to be periodic in φ and bounded as ε → 0. In the standard terminology of multiscale analysis, we would call x and φ the fast variables of these systems, y and I the slow variables. But we also have a particular interest on systems for which the fast and slow variables exist but cannot be explicitly identified beforehand. At the present time, there does not exist a unified strategy for dealing with both problems of type (1.1) and (1.2). There is, however, a large literature on stiff systems of the type (1.1) and oscillatory systems of the type (1.2) separately. Starting from the pioneering work of Dahlquist and Gear [5], there has been extensive work on designing efficient numerical methods for stiff ODEs [7], such as the backward differentiation formula, implicit Runge-Kutta methods [5], extrapolation methods and Rosenbrock methods [7, 8, 6]. There is also extensive work on oscillatory systems, some of which are analytical [1], and some are numerical [9]. We will apply the framework of the heterogeneous multiscale method (HMM) [3], which is a general methodology for problems with multiscales. In the case of two scales, a macroscale and a microscale, HMM consists of two components: selection of a conventional macroscale solver, here a standard ODE solver, and estimating the effective forces used in the macroscale solver by performing numerical experiments using the microscale model and processing the data obtained. The procedure can be iterated if the system has more than two separated scales. There are a number of related numerical methods for systems with multiple time scales, in particular for stiff ODEs. Even though the most popular numerical methods for stiff ODEs stem from implicit methods such as the backward differentiation formula, a number of explicit methods have also been proposed [8, 2, 7]. In the simplest version, these methods are Runge-Kutta in nature, each stage of which is a forward

The Riemann Problem for Fluid Flows in a Nozzle with Discontinuous Cross-Section

Abstract: The system of balance laws describing a compressible fluid flow in a nozzle forms a non-strictly hyperbolic system of partial differential equations which, also, is not fully conservative due to the effect of the geometry. First, we investigate the general properties of the system and determine all possible wave combinations. Second, we construct analytically the solutions of the Riemann problem for any values of the left-and right-handed states. For certain values we obtain up to three solution whose structure is carefully described here. In some range of Riemann data, no solutions exists. When three solutions are avialable, then exactly one of them contains two stationary waves which are superimposed in the physical space. We include also numerical plots of these solutions.

Computational high-frequency wave propogation using the level-set method with applications to the semi-classical limit of the Schrödinger equations

Abstract: We introduce a level set method for computational high frequency wave propagation in dispersive media and consider the application to linear Schrodinger equation with high frequency initial data. High frequency asymptotics of dispersive equations often lead to the well-known WKB system where the phase of the plane wave evolves according to a nonlinear Hamilton-Jacobi equation and the intensity is governed by a linear conservation law. From the Hamilton-Jacobi equation, wave fronts with multiple phases are constructed by solving a linear Liouville equation of a vector valued level set function in the phase space. The multi-valued phase itself can be constructed either from an additional linear hyperbolic equation in phase space or an additional linear homogeneous equation and component to the level set function in an augmented phase space. This phase is in fact valid in the entire physical domain, but one of the components of the level set function can be used to restrict it to a wave front of interest. The use of the level set method in this numerical approach provides an Eulerian framework that automatically resolves the multi-valued wave fronts and phase from the superposition of solutions of the equations in phase space.

Singularity Formation and Instability in the Unsteady Inviscid and Viscous Prandtl Equations

Abstract: We use the method of characteristics to prove the short-time existence of smooth solutions of the unsteady inviscid Prandtl equations, and present a simple explicit solution that forms a singularity in finite time. We give numerical and asymptotic solutions which indicate that this singularity persists for nonmonotone solutions of the viscous Prandtl equations. We also solve the linearization of the inviscid Prandtl equation about shear flow. We show that the resulting problem is weakly, but not strongly, well-posed, and that it has an unstable continuous spectrum when the shear flow has a critical point, in contrast with the behavior of the linearized Euler equations.

L-Stability of Constants in a Model for Radiating Gases

Abstract: In previous work, the L-stability of constant states in a model of radiative gases, under a zero-mass initial disturbance, was often left open. Actually, it was proved only for the Burgers flux and odd inital data which were non-negative on R+. We now prove this stability in full generality. This result is used, as usual, to prove the L-stability of shock profiles.

Effective One Particle Quantum Dynamics of Electrons: A Numerical Study of the Schrodinger-Poisson-X alpha Model

Abstract: The Schrodinger-Poisson-Xalpha (S-P-Xalpha) model is a "local one particle approximation" of the time dependent Hartree-Fock equations. It describes the time evolution of electrons in a quantum model respecting the Pauli principle in an approximate fashion which yields an effective potential that is the difference of the nonlocal Coulomb potential and the third root of the local density. We sketch the formal derivation, existence and uniqueness analysis of the S-P-Xalpha model with/without an external potential. In this paper we deal with numerical simulations based on a time-splitting spectral method, which was used and studied recently for the nonlinear Schrodinger (NLS) equation in the semi-classical regime and shows much better spatial and temporal resolution than finite difference methods. Extensive numerical results of position density an Winger measures in 1d, 2d, and 3d for the S-P-Xalpha model with/without an external potential are presented. These results give an insight to understand the interplay between the nonlocal ("weak") and the local ("strong") nonlinearity.

The 3D Quasigeostrophic Fluid Dynamics Under Random Forcing On Boundary

Abstract: The three-dimensional baroclinic quasigeostrophic flow model has been widely used to study basic mechanisms in oceanic flows and climate dynamics. In this paper, we consider this flow model under random wind forcing and time-periodic fluctuations on fluid boundary (the air-sea surface). The time-periodic fluctuations are due to periodic rotatios of the earth and thus periodic exposure of the earth to the solar radiation. After overcoming the difficulty due to the low regualrity of an associated Ornstein-Uhlenbeck process, we establish the well-posedness of the baroclinic quasigeostrophic flow model in the state space. Then we demonstrate the existence of the random attractors, again in the state space. We also discuss the relevance of out result to climate modeling.

Vlasov-Fokker-Plank Models for Multilane Traffic Flow

Abstract: Systems of equations of Vlasov-Fokker-Planck type are suggested for multilane traffic flow. The equations include nonlocal and time-delayed braking and acceleration terms with rates depending on the densities and relative speeds. The braking terms include lane-change probabilities. It is shown that simple natural assumptions on the structure of these probabilities lead to multivalued fundamental diagrams, consistent with traffic observations. Lane-changing behavior is the critical ingredient in such bifunctions.

Higher Order String Method for Finding Minimum Energy Paths

Abstract: The string method is an efficent numerical method for finding transition paths and transition rates in metastable systems. The dynamics of the string are governed by a Hamilton-Jacobi type of equation. We construct a stable and high order numerical scheme to estimate the first order spatial derivatives, or the tangent vectors in the equation. The construction is based on the idea of the upwind scheme and the essentially nonoscillatory scheme (ENO). Numerical examples demonstrate the improvement of the accuracy by the new scheme.

Incompressible Euler and E-MHD as scaling limits of the Vlasov-Maxwell system

Abstract: We consider two different asymptotic limits of the Vlasov-Maxwell system describing a quasineutral plasma with a uniform ionic background. In the first case, as the magnetic field is preserved in the limiting process, we obtain the so-called electron magnetohydrodynamics equations. In the second case, we obtain the incompressible Euler equations with no more magnetic field left.

A Fast Finite Differenc Method For Solving Navier-Stokes Equations on Irregular Domains

Abstract: A fast finite difference method is proposed to solve the incompressible Navier-Stokes equations defined on a general domain. The method is based on the voricity stream-function formulation and a fast Poisson solver defined on a general domain using the immersed interface method. The key to the new method is the fast Poisson solver for general domains and the interpolation scheme for the boundary condition of the stream function. Numerical examples thats show second order accuracy of the computed solution are also provided.

Convergence of the Spectral Method for Stochastic Ginzburg-Landau Equation Driven by Space-Time White Noise

Abstract: In this paper, a spectral method is formulated as a numerical solution for the stochastic Ginzburg-Landau equation driven by space-time white noise. The rates of pathwise convergence and convergence in expectation in Sobolev spaces are given based on the convergence rates of the spectral approximation for the stochastic convolution. The analysis can be generalized to other spectral methods for stochastic PDEs driven by additive noises, provided the regularity condition for the noises.

A Modeling of Compressible Droplets in a Fluid

Abstract: In this work, we are interested in a complex fluid-kinetic model that aims to take into account the compressibility of the droplets of the spray. The ambient fluid is described by Euler-like equations, in which the transfer of momentum and energy form the droplets is taken into account, while the spray is represented by a probability density function satisfying a Vlasov-like equation. Implicit terms crop up because of the compressibility of the droplets. After having derived the model, we prove that golbal conservations are satisfied. Then we present two numerical tests. The first one enables us to validate the numerical code, while the second one is performed in a physically realistic situation.

Generalized Navier Boundary Condition for the Moving Contact Line

Abstract: From molecular dynamics simulations on immiscible flows, we find the relative slipping between the fluids and the solid wall everywhere to follow the generalized Navier boundary condition, in which the amount of slipping is proportional to the sum of tangential viscous stress and the uncompensated Young stress. The latter arises from the deviation of the fluid-fluid interface from its static configuration. We give a continuum formulation of the immiscible flow hydrodynamics, comprising the generalized Navier boundary condition, the Navier-Stokes equation, and the Cahn- Hilliard interfacial free energy. Our hydrodynamic model yields near-complete slip of the contact line, with interfacial and velocity profiles matching quantitatively with those from the molecular dynamics simulations.

Adaptive Mesh Redistibution Method Based on Godunov's Scheme

Abstract: In this work, a detailed description for an efficent adaptive mesh redistribution algorithm based on the Godunov's scheme is presented. After each mesh iteration a second-order finite-volume flow solver is used to update the flow parameters at the new time level directly without using interpolation. Numerical experiments are perfomed to demonstrate the efficency and robustness of the proposed adaptive mesh algorithm in one and two dimensions.

Time Domain Computation of a Nonlinear Nonlocal Cochlear Model with Applications to Multitone Interaction in Hearing

Abstract: A nonlinear, nonlocal cochlear model of the transmission line type is studied in order to capture the multitone interactions and resulting tonal suppression effects. The model can serve as a module for voice signal processing, and is a one-dimensional (in space) damped dispersive nonlinear PDE based on the mechanics and phenomenology of hearing. It describes the motion of the basilar membrane (BM) in the cochlea driven by input pressure waves. Both elastic damping and selective longitudinal fluid damping are present. The former is nonlinear and nonlocal in BM displacement, and plays a key role in capturing tonal interactions. The latter is active only near the exit boundary (helicotrema), and is built in to damp out the remaining long waves. The initial boundary value problem is numerically solved with a semi-implicit second order finite difference method. Solutions reach a multi-frequency quasi-steady state. Numerical results are shown on two tone suppression from both high-frequency and low-frequency sides, consistent with known behavior of two tone suppression. Suppression effects among three tones are demonstrated by showing how the response magnitudes of the fixed two tones are reduced as we vary the third tone in frequency and amplitude. We observe qualitative agreement of our model solutions with existing cat auditory neural data. The model is thus a simple and efficient processing tool for voice signals.

Blowup of Solutions and Boundary Instabilities in Nonlinear Hyperbolic Equations

Abstract: We construct elementary examples of systems of hyperbolic equations having solutions which blow up in finite time. We explicitly describe the system, initial data and solution. First, we exhibit a 3x3 system with compactly supported data which blows up in finite time. The solutions blows up in amplitude (Linfinity] norm) on an entire interval, so there is no possibilty of continuing the solution beyond the blowup time. We then consider a system of two Burger equations which are coupled through linear boundary conditions. We record the interesting observation that although the IBVP with a single boundary condition is globally well-posed, when two boundary conditiond are used on a finite domain, the IBVP is ill-posed. Because waves are reflected back into the domain, multiple interactions combine to give blowup in finite time, for arbitrarily small initial data. We conclude that some global integral or energy condition must be imposed in order to expect stability of solutions to IBVPs on compact domains. Finally, we show that the presence of shocks is not necessary, by exhibiting solutions which are continuous in the nonlinear fields. However, our solutions do contain discontinuities in the linearly degenerate field.

Multi-Factor Financial Derivatives on Finite Domains

Abstract: In this paper, we introduce reversion conditions for stochastic models. Also we prove that if the models satisfy reversion conditions and the market prices of risks are bounded, then the final-value problem of general two-factor financial derivative equations on rectangular domains has a unique solution. For such problems we can obtain their numerical solutions without using any artificial conditions. Examples show that if the singularity-separating method and extrapolation techniques are used, then very good solutions can be obtained even on very coarse meshes.

On the Equation Satisfied by a Steady Prandtl-Munk Vortex Sheet

Abstract: We show the the voricity distribution obtained by minimizing the induced drag on a wing, the so called Prandtl-Munk vortex sheet, is not a travelling-wave weak solution of the Euler equations, contrary to what has been claimed by a number of authors. Instead, it is a weak solution of a non-homogeneous Euler equation, where the forcing term represents a "tension" force applied to the tips. This is consistent with a heuristic arguement due to Saffman. Thus, the notion of weak solution captures the correct physical behavior in this case.

Modeling 2D and 3D Horizontal Wells Using CVFA

Abstract: In this paper we present an application of the recently developed control volume function approximation (CVFA) method to the modeling and simulation of 2D and 3D horizontal wells in petroleum reservoirs. The base grid for this method is based on a Voronoi grid. One of the features of the CVFA is that the flux at the interfaces of control volumes can be accurately computed via function approximations. Also, it reduces grid orientation effects and applies to any shape of elements. It is particularly suitable for hybrid grid reservoir simulations. Through extensive numerical experiments and comparisons with the finite difference method for benchmark flow problems, we show that this method can effciently and accurately handle complex horizontal wells in any direction.

Nonpertubative Contributions in Quantum-Mechanical Models: The Instantonic Approach

Abstract: We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle. The configurations which allow construction of a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions. The quantum amplitudes consist of an exponential associated with the classical contribution as well as the energy eigenvalues of the quadratic operators at issue can be written in closed form due to the shape-invariance property. Accordingly, we resort to the zeta-function method to compute the functional determinants in a systematic way. The effect of the multi-instantons configurations is also carefully considered. To illustrate the instanton calculus in a relevant model, we go to the double-wall potential. The second popular case is the periodic-potential where the initial levels split into bands. The quantum decay rate of the metastable states in a cubic model is evaluated by means of the bounce-like solution.