614 research outputs found
Numerical investigations of traveling singular sources problems via moving mesh method
This paper studies the numerical solution of traveling singular sources
problems. In such problems, a big challenge is the sources move with different
speeds, which are described by some ordinary differential equations. A
predictor-corrector algorithm is presented to simulate the position of singular
sources. Then a moving mesh method in conjunction with domain decomposition is
derived for the underlying PDE. According to the positions of the sources, the
whole domain is splitted into several subdomains, where moving mesh equations
are solved respectively. On the resulting mesh, the computation of jump
is avoided and the discretization of the underlying PDE is reduced
into only two cases. In addition, the new method has a desired second-order of
the spatial convergence. Numerical examples are presented to illustrate the
convergence rates and the efficiency of the method. Blow-up phenomenon is also
investigated for various motions of the sources
Blowup in diffusion equations: A survey
AbstractThis paper deals with quasilinear reaction-diffusion equations for which a solution local in time exists. If the solution ceases to exist for some finite time, we say that it blows up. In contrast to linear equations blowup can occur even if the data are smooth and well-defined for all times. Depending on the equation either the solution or some of its derivatives become singular. We shall concentrate on those cases where the solution becomes unbounded in finite time. This can occur in quasilinear equations if the heat source is strong enough. There exist many theoretical studies on the question on the occurrence of blowup. In this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power. A better knowledge would be useful also for their numerical treatment. Thus, not surprisingly, the numerical analysis of this type of problems is still at a rather early stage. The goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems
Lagrangian and geometric analysis of finite-time Euler singularities
We present a numerical method of analyzing possibly singular incompressible
3D Euler flows using massively parallel high-resolution adaptively refined
numerical simulations up to 8192^3 mesh points. Geometrical properties of
Lagrangian vortex line segments are used in combination with analytical
non-blowup criteria by Deng et al [Commun. PDE 31 (2006)] to reliably
distinguish between singular and near-singular flow evolution. We then apply
the presented technique to a class of high-symmetry initial conditions and
present numerical evidence against the formation of a finite-time singularity
in this case.Comment: arXiv admin note: text overlap with arXiv:1210.253
Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods
The numerical simulation of the 3D incompressible Euler equation is analyzed
with respect to different integration methods. The numerical schemes we
considered include spectral methods with different strategies for dealiasing
and two variants of finite difference methods. Based on this comparison, a
Kida-Pelz like initial condition is integrated using adaptive mesh refinement
and estimates on the necessary numerical resolution are given. This estimate is
based on analyzing the scaling behavior similar to the procedure in critical
phenomena and present simulations are put into perspective.Comment: Euler equations: 250 years o
A moving mesh method with variable relaxation time
We propose a moving mesh adaptive approach for solving time-dependent partial
differential equations. The motion of spatial grid points is governed by a
moving mesh PDE (MMPDE) in which a mesh relaxation time \tau is employed as a
regularization parameter. Previously reported results on MMPDEs have invariably
employed a constant value of the parameter \tau. We extend this standard
approach by incorporating a variable relaxation time that is calculated
adaptively alongside the solution in order to regularize the mesh appropriately
throughout a computation. We focus on singular problems involving self-similar
blow-up to demonstrate the advantages of using a variable relaxation ime over a
fixed one in terms of accuracy, stability and efficiency.Comment: 21 page
Singularities of Euler flow? Not out of the blue!
Does three-dimensional incompressible Euler flow with smooth initial
conditions develop a singularity with infinite vorticity after a finite time?
This blowup problem is still open. After briefly reviewing what is known and
pointing out some of the difficulties, we propose to tackle this issue for the
class of flows having analytic initial data for which hypothetical real
singularities are preceded by singularities at complex locations. We present
some results concerning the nature of complex space singularities in two
dimensions and propose a new strategy for the numerical investigation of
blowup.(A version of the paper with higher-quality figures is available at
http://www.obs-nice.fr/etc7/complex.pdf)Comment: RevTeX4, 10 pages, 9 figures. J.Stat.Phys. in press (updated version
Structures and waves in a nonlinear heat-conducting medium
The paper is an overview of the main contributions of a Bulgarian team of
researchers to the problem of finding the possible structures and waves in the
open nonlinear heat conducting medium, described by a reaction-diffusion
equation. Being posed and actively worked out by the Russian school of A. A.
Samarskii and S.P. Kurdyumov since the seventies of the last century, this
problem still contains open and challenging questions.Comment: 23 pages, 13 figures, the final publication will appear in Springer
Proceedings in Mathematics and Statistics, Numerical Methods for PDEs:
Theory, Algorithms and their Application
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