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 \ud 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 \ud 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 \ud Key words and phrases: atomistic material models, quasicontinuum method, error analysis, adaptivity, stability\ud \ud 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 \ud 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