12,258 research outputs found
Large order Reynolds expansions for the Navier-Stokes equations
We consider the Cauchy problem for the incompressible homogeneous
Navier-Stokes (NS) equations on a d-dimensional torus, in the C^infinity
formulation described, e.g., in [25]. In [22][25] it was shown how to obtain
quantitative estimates on the exact solution of the NS Cauchy problem via the
"a posteriori" analysis of an approximate solution; such estimates concern the
interval of existence of the exact solution and its distance from the
approximate solution. In the present paper we consider an approximate solutions
of the NS Cauchy problem having the form u^N(t) = sum_{j=0}^N R^j u_j(t), where
R is the "mathematical" Reynolds number (the reciprocal of the kinematic
viscosity) and the coefficients u_j(t) are determined stipulating that the NS
equations be satisfied up to an error O(R^{N+1}). This subject was already
treated in [24], where, as an application, the Reynolds expansion of order N=5
in dimension d=3 was considered for the initial datum of Behr-Necas-Wu (BNW).
In the present paper, these results are enriched regarding both the theoretical
analysis and the applications. Concerning the theoretical aspect, we refine the
approach of [24] following [25] and use the symmetries of the initial datum in
building up the expansion. Concerning the applicative aspect we consider two
more (d=3) initial data, namely, the vortices of Taylor-Green (TG) and
Kida-Murakami (KM); the Reynolds expansions for the BNW, TG and KM data are
performed via a Python program, attaining orders between N=12 and N=20. Our a
posteriori analysis proves, amongst else, that the solution of the NS equations
with anyone of the above three data is global if R is below an explicitly
computed critical value. Our critical Reynolds numbers are below the ones
characterizing the turbulent regime; however these bounds have a sound
theoretical support, are fully quantitative and improve previous results of
global existence.Comment: Some overlaps with our works arXiv:1405.3421, arXiv:1310.5642,
arXiv:1304.2972, arXiv:1203.6865, arXiv:1104.3832, arXiv:1009.2051,
arXiv:1007.4412, arXiv:0909.3707, arXiv:0709.1670. These overlaps aim to make
the paper self-cointained and do not involve the main result
A plethora of generalised solitary gravity-capillary water waves
The present study describes, first, an efficient algorithm for computing
capillary-gravity solitary waves solutions of the irrotational Euler equations
with a free surface and, second, provides numerical evidences of the existence
of an infinite number of generalised solitary waves (solitary waves with
undamped oscillatory wings). Using conformal mapping, the unknown fluid domain,
which is to be determined, is mapped into a uniform strip of the complex plane.
In the transformed domain, a Babenko-like equation is then derived and solved
numerically.Comment: 20 pages, 7 figures, 45 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
A-posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension
The quasicontinuum (QC) method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we give an a-posteriori error analysis for the quasi-continuum method in one dimension. We consider atomistic models with Lennard-Jones type finite-range interactions.\ud
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We prove that, for a stable QC solution with a sufficiently small residual, which is computed in a discrete Sobolev-type norm, there exists an exact solution of the atomistic model problem for which an a-posteriori error estimate holds. We then derive practically computable bounds on the residual and on the inf-sup constants which measure the stability of the QC solution.\ud
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Finally, we supplement the QC method with a proximal point optimization method with local-error control. We prove that the parameters can be adjusted so that at each step of the optimization algorithm there exists an exact solution to a related atomistic problem whose distance to the numerical solution is smaller than a pre-set tolerance.\ud
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Key words and phrases: atomistic material models, quasicontinuum method, error analysis, adaptivity, stability\ud
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The first author acknowledges the financial support received from the European research project HPRB-CT-2002-00284: New Materials, Adaptive Systems and their Nonlinearities. Modelling, Control and Numerical Simulation, and the kind hospitality of Carlo Lovadina and Matteo Negri (University of Pavia).\ud
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We would like to thank Nick Gould for his advice on practical optimization methods, particularly on proximal point algorithms
Convergence of Adaptive Finite Element Approximations for Nonlinear Eigenvalue Problems
In this paper, we study an adaptive finite element method for a class of a
nonlinear eigenvalue problems that may be of nonconvex energy functional and
consider its applications to quantum chemistry. We prove the convergence of
adaptive finite element approximations and present several numerical examples
of micro-structure of matter calculations that support our theory.Comment: 24 pages, 12 figure
Computation of maximal local (un)stable manifold patches by the parameterization method
In this work we develop some automatic procedures for computing high order
polynomial expansions of local (un)stable manifolds for equilibria of
differential equations. Our method incorporates validated truncation error
bounds, and maximizes the size of the image of the polynomial approximation
relative to some specified constraints. More precisely we use that the manifold
computations depend heavily on the scalings of the eigenvectors: indeed we
study the precise effects of these scalings on the estimates which determine
the validated error bounds. This relationship between the eigenvector scalings
and the error estimates plays a central role in our automatic procedures. In
order to illustrate the utility of these methods we present several
applications, including visualization of invariant manifolds in the Lorenz and
FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable
manifolds in a suspension bridge problem. In the present work we treat
explicitly the case where the eigenvalues satisfy a certain non-resonance
condition.Comment: Revised version, typos corrected, references adde
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