5,903 research outputs found
A 8-neighbor model lattice Boltzmann method applied to mathematical-physical equations
© 2016. This version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A lattice Boltzmann method (LBM) 9-bit model is presented to solve mathematical-physical equations, such as, Laplace equation, Poisson equation, Wave equation and Burgers equation. The 9-bit model has been verified by several test cases. Numerical simulations, including 1D and 2D cases, of each problem are shown respectively. Comparisons are made between numerical predictions and analytic solutions or available numerical results from previous researchers. It turned out that the 9-bit model is computationally effective and accurate for all different mathematical-physical equations studied. The main benefits of the new model proposed is that it is faster than the previous existing models and has a better accuracy.Peer ReviewedPostprint (author's final draft
Fronts in randomly advected and heterogeneous media and nonuniversality of Burgers turbulence: Theory and numerics
A recently established mathematical equivalence--between weakly perturbed
Huygens fronts (e.g., flames in weak turbulence or geometrical-optics wave
fronts in slightly nonuniform media) and the inviscid limit of
white-noise-driven Burgers turbulence--motivates theoretical and numerical
estimates of Burgers-turbulence properties for specific types of white-in-time
forcing. Existing mathematical relations between Burgers turbulence and the
statistical mechanics of directed polymers, allowing use of the replica method,
are exploited to obtain systematic upper bounds on the Burgers energy density,
corresponding to the ground-state binding energy of the directed polymer and
the speedup of the Huygens front. The results are complementary to previous
studies of both Burgers turbulence and directed polymers, which have focused on
universal scaling properties instead of forcing-dependent parameters. The
upper-bound formula can be heuristically understood in terms of renormalization
of a different kind from that previously used in combustion models, and also
shows that the burning velocity of an idealized turbulent flame does not
diverge with increasing Reynolds number at fixed turbulence intensity, a
conclusion that applies even to strong turbulence. Numerical simulations of the
one-dimensional inviscid Burgers equation using a Lagrangian finite-element
method confirm that the theoretical upper bounds are sharp within about 15% for
various forcing spectra (corresponding to various two-dimensional random
media). These computations provide a new quantitative test of the replica
method. The inferred nonuniversality (spectrum dependence) of the front speedup
is of direct importance for combustion modeling.Comment: 20 pages, 2 figures, REVTeX 4. Moved some details to appendices,
added figure on numerical metho
The instanton method and its numerical implementation in fluid mechanics
A precise characterization of structures occurring in turbulent fluid flows
at high Reynolds numbers is one of the last open problems of classical physics.
In this review we discuss recent developments related to the application of
instanton methods to turbulence. Instantons are saddle point configurations of
the underlying path integrals. They are equivalent to minimizers of the related
Freidlin-Wentzell action and known to be able to characterize rare events in
such systems. While there is an impressive body of work concerning their
analytical description, this review focuses on the question on how to compute
these minimizers numerically. In a short introduction we present the relevant
mathematical and physical background before we discuss the stochastic Burgers
equation in detail. We present algorithms to compute instantons numerically by
an efficient solution of the corresponding Euler-Lagrange equations. A second
focus is the discussion of a recently developed numerical filtering technique
that allows to extract instantons from direct numerical simulations. In the
following we present modifications of the algorithms to make them efficient
when applied to two- or three-dimensional fluid dynamical problems. We
illustrate these ideas using the two-dimensional Burgers equation and the
three-dimensional Navier-Stokes equations
A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation
In this paper we analyze the large-time behavior of the augmented Burgers
equation. We first study the well-posedness of the Cauchy problem and obtain
- decay rates. The asymptotic behavior of the solution is obtained by
showing that the influence of the convolution term is the same as
for large times. Then, we propose a semi-discrete numerical scheme
that preserves this asymptotic behavior, by introducing two correcting factors
in the discretization of the non-local term. Numerical experiments illustrating
the accuracy of the results of the paper are also presented.Comment: Accepted for publication in ESAIM: Mathematical Modelling and
Numerical Analysi
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