Browse Prior Art Database

REGULARIZATION OF CONTOUR DYNAMICAL ALGORITHMS. I. TANGENTIAL REGUARIZATION

IP.com Disclosure Number: IPCOM000149504D
Original Publication Date: 1983-Dec-31
Included in the Prior Art Database: 2007-Apr-01
Document File: 40 page(s) / 1M

Publishing Venue

Software Patent Institute

Related People

Zabusky, Norman J.: AUTHOR [+3]

Abstract

Technical Report ICMA-82-48 REGULARIZATION OF CONTOUR DYNAMICAL ALGORITHMS. I. TANGENTIAL REGULARIZATION Norman J. Zabusky and Edward A. Overman, I1 Institute for Computational Mathematics and Applications Department of Mathematics and Statistics University of Pittsburgh Pittsburgh, PA 15261 ABSTRACT Contour dynamical methods are being applied to a variety of inviscid incompressible flows in two dimensions. These generalizations of the "waterbag" method provide simplified models for following the evolution of contours 5 (j) that separate regions of constant density which are the sources of the flow. The inviscid evolution of contour j, x is usually -t an area-preserving map. For physically unstable problems, a piecewise constant initial condition may result in an ill-posed problem. That is, contours may rapidly grow in perimeter and/or develop singularities and numerically induced small-scale structures in a finite time. To avoid such problems and model realistic weakly-dissipative or weakly dispersive flows, contour regularization procedures are required. In this paper we introduce

This text was extracted from a PDF file.
At least one non-text object (such as an image or picture) has been suppressed.
This is the abbreviated version, containing approximately 10% of the total text.

Page 1 of 40

Technical Report ICMA-82-48

REGULARIZATION OF CONTOUR DYNAMICAL ALGORITHMS.
I. TANGENTIAL REGULARIZATION

Norman J. Zabusky and Edward A. Overman, I1

Institute for Computational Mathematics and Applications
Department of Mathematics and Statistics

University of Pittsburgh

Pittsburgh, PA 15261

[This page contains 1 picture or other non-text object]

Page 2 of 40

ABSTRACT

    Contour dynamical methods are being applied to a variety of inviscid
incompressible flows in two dimensions. These generalizations of the
"waterbag" method provide simplified models for following the evolution of
contours 5 (j) that separate regions of constant density which are the
sources of the flow. The inviscid evolution of contour j, x is usually

-t

an area-preserving map. For physically unstable problems, a piecewise
constant initial condition may result in an ill-posed problem. That is,
contours may rapidly grow in perimeter and/or develop singularities and
numerically induced small-scale structures in a finite time. To avoid such
problems and model realistic weakly-dissipative or weakly dispersive flows,
contour regularization procedures are required. In this paper we introduce

,

dissipative and dispersive tangential regularization procedures for one

.

contour. A special case of the former, namely x = pzss, corresponds in

-t

lowest order to a linear diffusion operator in two dimensions. The contour
is parameterized with arc length using cubic splines and an adaptive
curvature controlled node insertion, removal and adjustment algorithm is used.
A modified Crank-Nicolson method is used to solve the discrete representation

h

of the full system, xt = gt + pxss. Numerical results are given for the
evolution of initially elliptical shapes according to prescribed area-
preserving maps. The numerical results for area evolution agree with
analytical results.

[This page contains 1 picture or other non-text object]

Page 3 of 40

1.2 Regularization Concepts for One and Two-Dimensional Flows

A cogent example of regularization is provided by Burgers' equation

an ideal model of one-dimensional pressureless hydrodynamics, where v > 0

is a constant viscous parameter. For bounded and smooth u (x), (1.1) has

0

unique solutions and bounded derivatives for all times [12,13]. However,
if one "simplifies" the problem by setting V = 0, the resultin─Łuler
equation

has the general solution

-1

and derivatives of u become singular at a time tg = lmin ulI . Thus, for

0

v > 0, characteristics are ''prevented" from crossing.

    Similarly, for computational studies of shock-wave problems in one-
dimension with negligibly small "true" viscosity, Richtmyer and von Neumann
[14,15J advocated the addition of a small-but-finite artificial or pseudo-
nonlinear viscosity to regularize the problem. In essence, they replaced the
pressure p by pfq, where

2

has units of (L/T) and GT is proportional to the...