José M. Vega

On the linearization of some singular, nonlinear elliptic problems and applications☆

This paper deals with the spectrum of a linear, weighted eigenvalue problem associated with a singular, second order, elliptic operator in a bounded domain, with Dirichlet boundary data. In particular, we analyze the existence and uniqueness of principal eigenvalues. As an application, we extend the usual concepts of linearization and Frechet derivability, and the method of sub and supersolutions to some semilinear, singular elliptic problems.

Reduced order models based on local POD plus Galerkin projection

A method is presented to accelerate numerical simulations on parabolic problems using a numerical code and a Galerkin system (obtained via POD plus Galerkin projection) on a sequence of interspersed intervals. The lengths of these intervals are chosen according to several basic ideas that include an a priori estimate of the error of the Galerkin approximation. Several improvements are introduced that reduce computational complexity and deal with: (a) updating the POD manifold (instead of calculating it) at the end of each Galerkin interval; (b) using only a limited number of mesh points to calculate the right hand side of the Galerkin system; and (c) introducing a second error estimate based on a second Galerkin system to account for situations in which qualitative changes in the dynamics occur during the application of the Galerkin system. The resulting method, called local POD plus Galerkin projection method, turns out to be both robust and efficient. For illustration, we consider a time-dependent Fisher-like equation and a complex Ginzburg-Landau equation.

Surface-wave damping in a brimful circular cylinder

As pointed out to us by Mr T. Heath, the following printing errors can be quite misleading when using the formulas in the paper to obtain eigenfrequencies and damping rates to compare with experiments: in (A 13) 1 should read −1 on the right-hand side; in (A 22) and (A 26) Ω20 should read Ω−20; in (A 25) the factor Ω40 must be omitted on the right-hand side. When revising again the printed version of the paper, we discovered several additional misprints: A factor C was omitted in the first two integrals in the expression for J2, immediately following equation (2.9). The sign of the second expression for I1 in (2.23) should be changed. The expression (W0Wz +3WW0z)z=0 should read 2(W0Wz +WW0z)z=0 in equation (2.24). The expression W0(1, z)W0z(1, z) in (2.26) should read W0(r, 0)W0z(r, 0). None of the misprints above affect the results of the paper, which were obtained with the correct expressions.

Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio

Nearly inviscid parametrically excited surface gravity–capillary waves in two-dimensional domains of finite depth and large aspect ratio are considered. Coupled equations describing the evolution of the amplitudes of resonant left- and right-traveling waves and their interaction with a mean flow in the bulk are derived, and the conditions for their validity established. Under suitable conditions the mean flow consists of an inviscid part together with a viscous mean flow driven by a tangential stress due to an oscillatory viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer. These forcing mechanisms are important even in the limit of vanishing viscosity, and provide boundary conditions for the Navier–Stokes equation satisfied by the mean flow in the bulk. For moderately large aspect ratio domains the amplitude equations are nonlocal but decouple from the equations describing the interaction of the slow spatial phase and the viscous mean flow. Two cases are considered in detail, gravity–capillary waves and capillary waves in a microgravity environment.

Local POD Plus Galerkin Projection in the Unsteady Lid-Driven Cavity Problem

A local proper orthogonal decomposition (POD) plus Galerkin projection method is applied to the unsteady lid-driven cavity problem, namely the incompressible fluid flow in a two-dimensional box whose upper wall is moved back and forth at moderately large values of the Reynolds number. Such a method was recently developed for one-dimensional parabolic problems. Its extension to fluid dynamics problems is nontrivial (especially if rough CFD codes are used) and consists of using a computational fluid dynamics (CFD) code and a Galerkin system (GS) in a sequence of interspersed intervals

Generation of Aerodynamics Databases Using High-Order Singular Value Decomposition

I_{CFD}

Linear oscillations of weakly dissipative axisymmetric liquid bridges

and

Linear oscillations of axisymmetric viscous liquid bridges

I_{GS}

On the steady streaming flow due to high-frequency vibration in nearly inviscid liquid bridges

, respectively. The POD manifold is calculated retaining the most energetic POD modes resulting from the snapshots computed in the

Dynamics of counterpropagating waves in parametrically forced systems

I_{CFD}