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Multiscale Modeling & Simulation

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2003

Volume 1, Issue 4, pp. 527-710


White Noise Limits for Inertial Particles in a Random Field

G. A. Pavliotis and A. M. Stuart

Multiscale Model. Simul. 1, pp. 527-553 (27 pages) | Cited 4 times

Online Publication Date: July 26, 2006

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In this paper we present a rigorous analysis of a scaling limit related to the motion of an inertial particle in a Gaussian random field. The mathematical model comprises Stokes's law for the particle motion and an infinite dimensional Ornstein-Uhlenbeck process for the fluid velocity field. The scaling limit studied leads to a white noise limit for the fluid velocity, which balances particle inertia and the friction term. Strong convergence methods are used to justify the limiting equations. The rigorously derived limiting equations are of physical interest for the concrete problem under investigation and facilitate the study of two-point motions in the white noise limit. Furthermore, the methodology developed may also prove useful in the study of various other asymptotic problems for stochastic differential equations in infinite dimensions.

Reaction-Diffusion Front Speeds in Spatially-Temporally Periodic Shear Flows

Jim Nolen and Jack Xin

Multiscale Model. Simul. 1, pp. 554-570 (17 pages) | Cited 6 times

Online Publication Date: July 26, 2006

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We study the asymptotics of two space dimensional reaction-diffusion front speeds through mean zero space-time periodic shears using both analytical and numerical methods. The analysis hinges on traveling fronts and their estimates based on qualitative properties such as monotonicity and a priori integral inequalities. The computation uses an explicit second order upwind finite difference method to provide more quantitative information. At small shear amplitudes, front speeds are enhanced by an amount proportional to shear amplitude squared. The proportionality constant has a closed form expression. It decreases with increasing shear temporal frequency and is independent of the form of the known reaction nonlinearities. At large shear amplitudes and for all reaction nonlinearities, the enhanced speeds grow proportional to shear amplitude and are again decreasing with increasing shear temporal frequencies. The results extend previous ones in the literature on front speeds through spatially periodic shears and show front speed slowdown due to shear direction switching in time.

A Rate-Independent Model for Inelastic Behavior of Shape-Memory Alloys

Alexander Mielke and Tomás Roubícek

Multiscale Model. Simul. 1, pp. 571-597 (27 pages) | Cited 19 times

Online Publication Date: July 26, 2006

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We formulate a model describing rate-independent hysteretic response of shape-memory alloys under slow external forcing. Under natural assumptions we prove that this model has a solution. The microstructure is treated on a "mesoscopic" level, described by volume fractions of particular phases in terms of Young measures. The whole formulation is based on energetic functionals for energy storage and energy dissipation. The latter is built into the model by a dissipation distance between different values of these volume fractions.

Eulerian Moment Equations for 2-D Stochastic Immiscible Flow

Kenneth D. Jarman and Thomas F. Russell

Multiscale Model. Simul. 1, pp. 598-608 (11 pages) | Cited 9 times

Online Publication Date: July 26, 2006

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We solve statistical moment differential equations (MDEs) for immiscible flow in porous media in the limit of zero capillary pressure, with application to secondary oil recovery. Closure is achieved by Taylor expansion of the fractional flow function and a perturbation argument. Previous results in one dimension are extended to two dimensions. Comparison to Monte Carlo simulation (MCS) shows that the MDE approach gives a good approximation to total oil production. For such spatially integrated or averaged quantities MDEs may be substantially more efficient than MCS.

Time-Reversed Refocusing of Surface Water Waves

Jean-Pierre Fouque and André Nachbin

Multiscale Model. Simul. 1, pp. 609-629 (21 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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A time-reversal mirror is, roughly speaking, a device which is capable of receiving a signal in time, keeping it in memory, and sending it back into the medium in the reversed direction of time. A brief mathematical review of the time-reversal (in reflection) theory is presented in the context of the linear shallow water equations. In particular, an explicit expression is given for the refocused pulse in the simplest time-reversal case. The explicit expression for the power spectral density of the reflection process is used to construct the highpass filter, which controls the refocusing process. Time-reversal numerical experiments in the (effectively) linear regime are used to validate the nonlinear shallow water code. The numerically refocused pulse is compared with the theoretical predicted shape. Further numerical experiments illustrate the robustness of the theory, in particular the time-reversal refocusing with smaller cutoff windows, the self-averaging property, and finally refocusing when the nonlinear term is small but not negligible.

Renormalization Group Analysis of Nonlinear Diffusion Equations with Periodic Coefficients

Gastão A. Braga, Frederico Furtado, Jussara M. Moreira, and Leonardo T. Rolla

Multiscale Model. Simul. 1, pp. 630-644 (15 pages)

Online Publication Date: July 26, 2006

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In this paper we present an efficient numerical approach based on the renormalization group method for the computation of self-similar dynamics. The latter arise, for instance, as the long-time asymptotic behavior of solutions to nonlinear parabolic partial differential equations. We illustrate the approach with the verification of a conjecture about the long-time behavior of solutions to a certain class of nonlinear diffusion equations with periodic coefficients. This conjecture is based on a mixed argument involving ideas from homogenization theory and the renormalization group method. Our numerical approach provides a detailed picture of the asymptotics, including the determination of the effective or renormalized diffusion coefficient.

Drawing Huge Graphs by Algebraic Multigrid Optimization

Yehuda Koren, Liran Carmel, and David Harel

Multiscale Model. Simul. 1, pp. 645-673 (29 pages) | Cited 5 times

Online Publication Date: July 26, 2006

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We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits a vast improvement over the fastest algorithms we are currently aware of; using a serial PC, it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. The minimization problem is expressed as a generalized eigenvalue problem, which is solved rapidly using a novel algebraic multigrid technique. The same generalized eigenvalue problem seems to come up also in other fields; hence ACE appears to be applicable outside graph drawing too.

Semiclassical Wigner Function and Geometrical Optics

Stathis Filippas and George N. Makrakis

Multiscale Model. Simul. 1, pp. 674-710 (37 pages) | Cited 8 times

Online Publication Date: July 26, 2006

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We consider the problem of high-frequency asymptotics for the time-dependent one-dimensional Schrödinger equation with rapidly oscillating initial data. This problem is commonly studied via the WKB method. An alternative method is based on the limit Wigner measure. This approach recovers geometrical optics, but, like the WKB method, it fails at caustics. To remedy this deficiency we employ the semiclassical Wigner function which is a formal asymptotic approximation of the scaled Wigner function but also a regularization of the limit Wigner measure. We obtain Airy-type asymptotics for the semiclassical Wigner function. This representation is shown to be exact in the context of concrete examples. In these examples we compute both the semiclassical and the limit Wigner function, as well as the amplitude of the wave field near a fold or a cusp caustic, which evolve naturally from suitable initial data.
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