7,325 research outputs found
Spectral methods for partial differential equations
Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid dynamical applications are emphasized
The spt-Crank for Ordinary Partitions
The spt-function was introduced by Andrews as the weighted counting
of partitions of with respect to the number of occurrences of the smallest
part. Andrews, Garvan and Liang defined the spt-crank of an -partition which
leads to combinatorial interpretations of the congruences of mod 5 and
7. Let denote the net number of -partitions of with spt-crank
. Andrews, Garvan and Liang showed that is nonnegative for all
integers and positive integers , and they asked the question of finding
a combinatorial interpretation of . In this paper, we introduce the
structure of doubly marked partitions and define the spt-crank of a doubly
marked partition. We show that can be interpreted as the number of
doubly marked partitions of with spt-crank . Moreover, we establish a
bijection between marked partitions of and doubly marked partitions of .
A marked partition is defined by Andrews, Dyson and Rhoades as a partition with
exactly one of the smallest parts marked. They consider it a challenge to find
a definition of the spt-crank of a marked partition so that the set of marked
partitions of and can be divided into five and seven equinumerous
classes. The definition of spt-crank for doubly marked partitions and the
bijection between the marked partitions and doubly marked partitions leads to a
solution to the problem of Andrews, Dyson and Rhoades.Comment: 22 pages, 6 figure
Spectral multigrid methods with applications to transonic potential flow
Spectral multigrid methods are demonstrated to be a competitive technique for solving the transonic potential flow equation. The spectral discretization, the relaxation scheme, and the multigrid techniques are described in detail. Significant departures from current approaches are first illustrated on several linear problems. The principal applications and examples, however, are for compressible potential flow. These examples include the relatively challenging case of supercritical flow over a lifting airfoil
Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions
The second-grade fluid equations are a model for viscoelastic fluids, with
two parameters: , corresponding to the elastic response, and , corresponding to viscosity. Formally setting these parameters to
reduces the equations to the incompressible Euler equations of ideal fluid
flow. In this article we study the limits of solutions of
the second-grade fluid system, in a smooth, bounded, two-dimensional domain
with no-slip boundary conditions. This class of problems interpolates between
the Euler- model (), for which the authors recently proved
convergence to the solution of the incompressible Euler equations, and the
Navier-Stokes case (), for which the vanishing viscosity limit is
an important open problem. We prove three results. First, we establish
convergence of the solutions of the second-grade model to those of the Euler
equations provided , as , extending
the main result in [19]. Second, we prove equivalence between convergence (of
the second-grade fluid equations to the Euler equations) and vanishing of the
energy dissipation in a suitably thin region near the boundary, in the
asymptotic regime ,
as . This amounts to a convergence criterion similar to the
well-known Kato criterion for the vanishing viscosity limit of the
Navier-Stokes equations to the Euler equations. Finally, we obtain an extension
of Kato's classical criterion to the second-grade fluid model, valid if , as . The proof of all these results
relies on energy estimates and boundary correctors, following the original idea
by Kato.Comment: 20pages,1figur
On the subgrid-scale modeling of compressible turbulence
A subgrid-scale model recently derived for use in the large-eddy simulation of compressible turbulent flows is examined from a fundamental theoretical and computational standpoint. It is demonstrated that this model, which is applicable only to compressible turbulent flows in the limit of small density fluctuations, correlates somewhat poorly with the results of direct numerical simulations of compressible isotropic turbulence at low Mach numbers. An alternative model, based on Favre-filtered fields, is suggested which appears to reduce these limitations
Convergence of the 2D Euler- to Euler equations in the Dirichlet case: indifference to boundary layers
In this article we consider the Euler- system as a regularization of
the incompressible Euler equations in a smooth, two-dimensional, bounded
domain. For the limiting Euler system we consider the usual non-penetration
boundary condition, while, for the Euler- regularization, we use
velocity vanishing at the boundary. We also assume that the initial velocities
for the Euler- system approximate, in a suitable sense, as the
regularization parameter , the initial velocity for the limiting
Euler system. For small values of , this situation leads to a boundary
layer, which is the main concern of this work. Our main result is that, under
appropriate regularity assumptions, and despite the presence of this boundary
layer, the solutions of the Euler- system converge, as ,
to the corresponding solution of the Euler equations, in in space,
uniformly in time. We also present an example involving parallel flows, in
order to illustrate the indifference to the boundary layer of the limit, which underlies our work.Comment: 22page
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