17,599 research outputs found
Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems
Stochastic physical problems governed by nonlinear conservation laws are
challenging due to solution discontinuities in stochastic and physical space.
In this paper, we present a level set method to track discontinuities in
stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed
function that vanishes at discontinuities, the iso-zero of the level set
problem coincide with the discontinuities of the conservation law. The level
set problem is solved on a sequence of successively finer grids in stochastic
space. The method is adaptive in the sense that costly evaluations of the
conservation law of interest are only performed in the vicinity of the
discontinuities during the refinement stage. In regions of stochastic space
where the solution is smooth, a surrogate method replaces expensive evaluations
of the conservation law. The proposed method is tested in conjunction with
different sets of localized orthogonal basis functions on simplex elements, as
well as frames based on piecewise polynomials conforming to the level set
function. The performance of the proposed method is compared to existing
adaptive multi-element generalized polynomial chaos methods
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A numerical study of heat flow in an elliptic cylinder with interior derivative boundary conditions
A study of numerical methods for hyperbolic conservation laws with stiff source terms
The proper modeling of nonequilibrium gas dynamics is required in certain regimes of hypersonic flow. For inviscid flow this gives a system of conservation laws coupled with source terms representing the chemistry. Often a wide range of time scales is present in the problem, leading to numerical difficulties as in stiff systems of ordinary differential equations. Stability can be achieved by using implicit methods, but other numerical difficulties are observed. The behavior of typical numerical methods on a simple advection equation with a parameter-dependent source term was studied. Two approaches to incorporate the source term were utilized: MacCormack type predictor-corrector methods with flux limiters, and splitting methods in which the fluid dynamics and chemistry are handled in separate steps. Various comparisons over a wide range of parameter values were made. In the stiff case where the solution contains discontinuities, incorrect numerical propagation speeds are observed with all of the methods considered. This phenomenon is studied and explained
Double-slit and electromagnetic models to complete quantum mechanics
We analyze a realistic microscopic model for electronic scattering with the
neutral differential delay equations of motion of point charges of the
Wheeler-Feynman electrodynamics. We propose a microscopic model according to
the electrodynamics of point charges, complex enough to describe the essential
physics. Our microscopic model reaches a simple qualitative agreement with the
experimental results as regards interference in double-slit scattering and in
electronic scattering by crystals. We discuss our model in the light of
existing experimental results, including a qualitative disagreement found for
the double-slit experiment. We discuss an approximation for the complex neutral
differential delay equations of our model using piecewise-defined
(discontinuous) velocities for all charges and piecewise-constant-velocities
for the scattered charge. Our approximation predicts the De Broglie wavelength
as an inverse function of the incoming velocity and in the correct order of
magnitude. We explain the scattering by crystals in the light of the same
simplified modeling with Einstein-local interactions. We include a discussion
of the qualitative properties of the neutral-delay-equations of electrodynamics
to stimulate future experimental tests on the possibility to complete quantum
mechanics with electromagnetic models.Comment: 4 figures, the same post-publication typos over the published version
of Journal of Computational and Theoretical Nanoscience, only that these
correction are not marked in red as in V7, this one is for a recollectio
A delay differential model of ENSO variability: Parametric instability and the distribution of extremes
We consider a delay differential equation (DDE) model for El-Nino Southern
Oscillation (ENSO) variability. The model combines two key mechanisms that
participate in ENSO dynamics: delayed negative feedback and seasonal forcing.
We perform stability analyses of the model in the three-dimensional space of
its physically relevant parameters. Our results illustrate the role of these
three parameters: strength of seasonal forcing , atmosphere-ocean coupling
, and propagation period of oceanic waves across the Tropical
Pacific. Two regimes of variability, stable and unstable, are separated by a
sharp neutral curve in the plane at constant . The detailed
structure of the neutral curve becomes very irregular and possibly fractal,
while individual trajectories within the unstable region become highly complex
and possibly chaotic, as the atmosphere-ocean coupling increases. In
the unstable regime, spontaneous transitions occur in the mean ``temperature''
({\it i.e.}, thermocline depth), period, and extreme annual values, for purely
periodic, seasonal forcing. The model reproduces the Devil's bleachers
characterizing other ENSO models, such as nonlinear, coupled systems of partial
differential equations; some of the features of this behavior have been
documented in general circulation models, as well as in observations. We
expect, therefore, similar behavior in much more detailed and realistic models,
where it is harder to describe its causes as completely.Comment: 22 pages, 9 figure
From discretization to regularization of composite discontinuous functions
Discontinuities between distinct regions, described by different equation sets, cause difficulties for PDE/ODE solvers. We present a new algorithm that eliminates integrator discontinuities through regularizing discontinuities. First, the algorithm determines the optimum switch point between two functions spanning adjacent or overlapping domains. The optimum switch point is determined by searching for a “jump point” that minimizes a discontinuity between adjacent/overlapping functions. Then, discontinuity is resolved using an interpolating polynomial that joins the two discontinuous functions.
This approach eliminates the need for conventional integrators to either discretize and then link discontinuities through generating interpolating polynomials based on state variables or to reinitialize state variables when discontinuities are detected in an ODE/DAE system. In contrast to conventional approaches that handle discontinuities at the state variable level only, the new approach tackles discontinuity at both state variable and the constitutive equations level. Thus, this approach eliminates errors associated with interpolating polynomials generated at a state variable level for discontinuities occurring in the constitutive equations.
Computer memory space requirements for this approach exponentially increase with the dimension of the discontinuous function hence there will be limitations for functions with relatively high dimensions. Memory availability continues to increase with price decreasing so this is not expected to be a major limitation
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