40 research outputs found
Analysis of a Monte Carlo boundary propagation method
AbstractA modified Monte Carlo technique, first developed in estimating a solution to Poisson's equation, is described and estimates of its computational complexities are derived. The method yields better estimates than the standard Monte Carlo approach by incorporating boundary information more efficiently and by the implicit reuse of random walk information gathered throughout the course of the computation. The new approach reduces the computational complexity of the length of a random walk by one order of magnitude as compared to a standard method described in many text books. Also, the number of walks necessary to achieve a desired accuracy is reduced
Some qualitative properties of the solutions of the Magnetohydrodynamic equations for nonlinear bipolar fluids
In this article we study the long-time behaviour of a system of nonlinear
Partial Differential Equations (PDEs) modelling the motion of incompressible,
isothermal and conducting modified bipolar fluids in presence of magnetic
field. We mainly prove the existence of a global attractor denoted by \A for
the nonlinear semigroup associated to the aforementioned systems of nonlinear
PDEs. We also show that this nonlinear semigroup is uniformly differentiable on
\A. This fact enables us to go further and prove that the attractor \A is
of finite-dimensional and we give an explicit bounds for its Hausdorff and
fractal dimensions.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s10440-014-9964-
Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgersâ equation
In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgersâ equation in one space dimension. The a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredientsâboth of which admit efficient Offline-Online treatment: the first is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential-in-time stability factor. These error bounds serve both Offline for construction of the reduced basis space by a new POD-Greedy procedure and Online for verification of fidelity. The a posteriori error bounds are practicable for final times (measured in convective units) TâO(1) and Reynolds numbers Îœ[superscript â1]â«1; we present numerical results for a (stationary) steepening front for T=2 and 1â€Îœ[superscript â1]â€200.United States. Air Force Office of Scientific Research (AFOSR Grant FA9550-05-1-0114)United States. Air Force Office of Scientific Research (AFOSR Grant FA-9550-07-1-0425)Singapore-MIT Alliance for Research and Technolog
The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius
We consider the two-dimensional motion of the coupled system of a viscous
incompressible fluid and a rigid disc moving with the fluid, in the whole
plane. The fluid motion is described by the Navier-Stokes equations and the
motion of the rigid body by conservation laws of linear and angular momentum.
We show that, assuming that the rigid disc is not allowed to rotate, as the
radius of the disc goes to zero, the solution of this system converges, in an
appropriate sense, to the solution of the Navier-Stokes equations describing
the motion of only fluid in the whole plane. We also prove that the trajectory
of the centre of the disc, at the zero limit of its radius, coincides with a
fluid particle trajectory.Comment: 29 pages, 0 figure
Regularity issues in the problem of fluid structure interaction
We investigate the evolution of rigid bodies in a viscous incompressible
fluid. The flow is governed by the 2D Navier-Stokes equations, set in a bounded
domain with Dirichlet boundary conditions. The boundaries of the solids and the
domain have H\"older regularity , . First, we
show the existence and uniqueness of strong solutions up to collision. A key
ingredient is a BMO bound on the velocity gradient, which substitutes to the
standard estimate for smoother domains. Then, we study the asymptotic
behaviour of one body falling over a flat surface. We show that
collision is possible in finite time if and only if
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Analyses, algorithms, and computations for models of high-temperature superconductivity. Final technical report
Under the sponsorship of the Department of Energy, the authors have achieved significant progress in the modeling, analysis, and computation of superconducting phenomena. Their work has focused on mezoscale models as typified by the celebrated ginzburg-Landau equations; these models are intermediate between the microscopic models (that can be used to understand the basic structure of superconductors and of the atomic and sub-atomic behavior of these materials) and the macroscale, or homogenized, models (that can be of use for the design of devices). The models the authors have considered include a time dependent Ginzburg-Landau model, a variable thickness thin film model, models for high values of the Ginzburg-Landau parameter, models that account for normal inclusions and fluctuations and Josephson effects, and the anisotropic Ginzburg-Landau and Lawrence-Doniach models for layered superconductors, including those with high critical temperatures. In each case, they have developed or refined the models, derived rigorous mathematical results that enhance the state of understanding of the models and their solutions, and developed, analyzed, and implemented finite element algorithms for the approximate solution of the model equations