Oliver A. McBryan

Hypercube algorithms and implementations

Parallel algorithms are presented for important components of computational fluid dynamics algorithms along with implementations on hypercube computers. These programs, used to solve hyperbolic and...

The bifurcation of tracked scalar waves

The dynamic evolution of tracked waves by a front-tracking algorithm may lead on either numerical or physical grounds to intersections of the waves. The correct resolution of these intersections is described locally by the solution of Riemann problems and requires a bifurcation of the topology defined by the tracked waves. An algorithm is described which is appropriate for the resolution of scalar tracked waves, such as material discontinuities, contact dicontinuities in gas dynamics, or constituent concetration waves including oil-water banks in oil reservoirs Even here the algorithm is not fully general, and the resolution of the intersections of an arbitrary set of curves in the plane for the above range of physical problems remains unsolved. However with the assumption that the set of intersections to be resolved is a small perturbation (resulting for example from a small time step in an evolution) of a valid, non-intersecting front, the algorithm seems to be general. In any case examples will be presented that show that complicated interfaces can be generated automatically from simple ones through successive bifurcations. 15 refs., 9 figs.

Front tracking applied to Rayleigh Taylor instability

A numerical solution of the two-fluid incompressible Euler equation is used to study the Rayleigh–Taylor instability. The solution is based on the method of front tracking, which has the distinguishing feature of being a predominantly Eulerian method in which sharp interfaces are preserved with zero numerical diffusion. In this paper validation of the method is obtained by comparison with existing numerical solutions based on conformal mapping. An initial study of heterogeneity is presented.

A numerical method for two phase flow with an unstable interface

Abstract The random choice method is used to compute the oil-water interface for two dimensional porous media equations. The equations used are a pair of coupled equations: the (elliptic) pressure equation and the (hyperbolic) saturation equation. The equations do not include the dispersive capillary pressure term and the computation does not introduce numerical diffusion. The method resolves saturation discontinuities sharply. The main conclusion of this paper is that the random choice is a correct numerical procedure for this problem even in the highly fingered case. Two methods of inducing fingers are considered: deterministically, through choice of Cauchy data and heterogeneity, through maximizing the randomness of, the random choice method.

On the decay of correlations inSO(n)-symmetric ferromagnets

We prove that for low temperaturesT the spin-spin correlation function of the two-dimensional classicalSO(n)-symmetric Ising ferromagnet decays faster than |x|−constT providedn≧2. We also discuss a nearest neighbor continuous spin model, with spins restricted to a finite interval, where we show that the spin-spin correlation function decays exponentially in any number of dimensions.

A computational model for interfaces

Decompositions of the plane into disjoint components separated by curves occur frequently. We describe a package of subroutines which provides facilities for defining, building, and modifying such decompositions and for efficiently solving various point and area location problems. Beyond the point that the specification of this package may be useful to others, we reach the broader conclusion that well-designed data structures and support routines allow the use of more conceptual or non-numerical portions of mathematics in the computational process, thereby extending greatly the potential scope of the use of computers in scientific problem solving. Ideas from conceptual mathematics, symbolic computation, and computer science can be utilized within the framework of scientific computing and have an important role to play in that area.

Polymer floods: a case study of nonlinear wave analysis and of instability control in tertiary oil recovery

Polymer flooding in oil reservoir simulation is considered in two space dimensions. The wave structures associated with such a process give rise to interesting phenomena in the nonlinear regime which have direct bearing on the efficiency of oil recovery. These waves influence and can prevent surface instabilities of the fingering mode. In this paper we resolve these waves by a front tracking method. We consider the fingering problem and the issue of oil recovery for the polymer flood. The details of these two phenomena depend on the separation between the waves and upon the viscosity contrast between the oil, water and polymer. We identify a nonlinear transfer of instability between adjacent waves and a nonlinear enhancement of recovery due to successive waves. The conclusions produced by this work are also pertinent to tracer flooding.One interesting conclusion applies to polymer injection followed by pure water injection. In this case the instability is transferred to the polymer-water interface, and th...

Statistical fluid dynamics: unstable fingers

This paper is the first in a series by the authors devoted to the study of fingers in fluid surfaces. Fingers are a form of surface instability which occur on many length scales. In particular, they may occur on length scales small relative to the natural dimensions of the problem; in this sense the instability is similar to turbulance. In the present study, the transition from stability to instability is determined by a critical value in a viscosity ratio. This series of papers is devoted to methods of accurate numerical computation. We find that the random choice method gives excellent resolution of fingered surfaces and discontinuities. Even an unstable interface, with three to four well developed fingers can be resolved on a coarse grid of 10 to 15 zones wide.

Normalized Convergence Rates for the PSMG Method

In a previous paper [“Parallel Superconvergent Multigrid,” in Multigrid Methods, Marcel Dekker, New York, 1988] the authors introduced an efficient multiscale PDE solver for massively parallel architectures, which was called Parallel Superconvergent Multigrid, or PSMG. In this paper, sharp estimates are derived for the normalized work involved in PSMG solution—the number of parallel arithmetic and communication operations required per digit of error reduction. PSMG is shown to provide fourth-order accurate solutions of Poisson-type equations at convergence rates of .00165 per single relaxation iteration, and with parallel operation counts per grid level of 5.75 communications and 8.62 computations for each digit of error reduction. The authors show that PSMG requires less than one-half as many arithmetic and one-fifth as many communication operations, per digit of error reduction, as a parallel standard multigrid algorithm (RBTRB) presented recently by Decker [SIAM J. Sci. Statist. Comput., 12 (1991), pp. 208–220].

The ϕ 2 4 quantum field as a limit of Sine-Gordon fields

AbstractWe exhibit the λϕ24 quantum field theory as the limit of Sine-Gordon fields as suggested by the identity