John C. Strikwerda

A nonreflecting outflow boundary condition for subsonic Navier-Stokes calculations

Abstract A nonreflecting boundary condition is presented for the numerical solution of the time-dependent compressible Navier-Stokes equations when these equations are used to obtain a steady state. This boundary condition is shown to be effective in reducing reflections at a subsonic outflow boundary. Numerical calculations using a model problem were made to compare this boundary condition with other outflow boundary conditions. The non-reflecting boundary condition contains a parameter whose optimal value is estimated using the analysis of a simplified set of equations.

Boundary conditions for subsonic compressible Navier-Stokes calculations

A systematic study of inflow and outflow boundary conditions for the numerical solution of the compressible Navier-Stokes equations is presented. Combinations of several representative inflow and outflow boundary conditions are applied in the solution of subsonic flow over a flat plate in a finite computational domain. These boundary conditions are evaluated in terms of their effect on the accuracy of the solution and the rate of convergence to a steady state. It is shown that errors in the data specified at the inflow boundary can produce significant errors in the computed flow field. It is also shown that a non-reflecting outflow boundary condition can significantly reduce the total computational time required.

Finite Difference Methods for the Stokes and Navier–Stokes Equations

This paper presents a new finite difference scheme for the Stokes equations and incompressible Navier–Stokes equations for low Reynolds number. The scheme uses the primitive variable formulation of the equations and is applicable with nonuniform grids and nonrectangular geometries. Several other methods of solving the Navier–Stokes equations are also examined in this paper and some of their strengths and weaknesses are described. Computational results using the new scheme are presented for the Stokes equations for a region with curved boundaries and for a disk with polar coordinates. The results show the method to be second-order accurate.

The Accuracy of the Fractional Step Method

We analyze the accuracy of the fractional step method of Kim and Moin [J. Comput. Phys., 59 (1985), pp. 308--323] for the incompressible Navier--Stokes equations. We show that the boundary conditions cannot be exactly satisfied in the projection step and that this limits the accuracy of the method. We also show that the pressure in any projection method can be at best first-order accurate. Our analysis is simpler and more direct than the previous analyses of this method. We also show that there is no numerical boundary layer for velocity or pressure, but there is one for the auxiliary pressure variable.

Analysis of dynamic congestion control protocols: a Fokker-Planck approximation

We present an approximate analysis of a queue with dynamically changing input rates that are based on implicit or explicit feedback. This is motivated by recent proposals for adaptive congestion control algorithms [RaJa 88, Jac 88], where the senders window size at the transport level is adjusted based on perceived congestion level of a bottleneck node. We develop an analysis methodology for a simplified system; yet it is powerful enough to answer the important questions regarding stability, convergence (or oscillations), fairness and the significant effect that delayed feedback plays on performance. Specifically, we find that, in the absence of feedback delay, the linear increase/exponential decrease algorithm of Jacobson and Ramakrishnan-Jain [ Jac 88, RaJa 88] is provably stable and fair. Delayed feedback on the other hand, introduces oscillations for every individual user as well as unfairness across those competing for the same resource. While the simulation study of Zhang [Zha 89] and the fluid-approximation study of Bolot and Shanker [BoSh 90] have observed the oscillations in cumulative queue length and measurements by Jacobson [ Jac 88] have revealed some of the unfairness properties, the reasons for these have not been identified. We identify quantitatively the cause of these effects, via-a-vis the systems parameters and properties of the algorithm used. The model presented is fairly general and can be applied to evaluate the performance of a wide range of feedback control schemes. It is an extension of the classical Fokker-Planck equation. Therefore, it addresses traffic viability (to some extent) that fluid approximation techniques do not address. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-91-18. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/385 Analysis Of Dynamic Congestion Control Protocols: A Fokker-Planck Approximation MS-CIS-91-18 DISTRIBUTED SYSTEMS LAB 5 Amarnath Mukherjee John C. Strikwerda Department of Computer and Information Science School of Engineering and Applied Science University of Pennsylvania Philadelphia, PA 19104-6389

Initial boundary value problems for the method of lines

This paper treats the stability of the initial boundary value problem for the method of lines applied to hyperbolic and parabolic partial differential equations in one space dimension. The theory treats the case of variable coefficients and allows for very general boundary conditions. Several examples are given which illustrate the theory. The theory is analogous to that developed by Gustafsson, Kreiss, and Sundstrom for finite-difference methods.

Interior Regularity Estimates for Elliptic Systems of Difference Equations

We prove interior regularity estimates for a large class of difference approximations for elliptic systems of partial differential equations. These estimates are analogous to those for the systems of differential equations. From these regularity estimates we obtain estimates on the convergence of the solutions of the difference equations. A theory of pseudo-difference operators with order is used to prove the regularity results. We also comment on factors to be considered in choosing a difference scheme for computations.

A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions

We survey results related to the Kreiss Matrix Theorem, especially examining extensions of this theorem to Banach space and Hilbert space. The survey includes recent and established results together with proofs of many of the interesting facts concerning the Kreiss Matrix Theorem.

A probabilistic analysis of asynchronous iteration

Abstract We present and analyze a probabilistic model for asynchronous iteration of linear systems. The model is similar in spirit to the chaotic model proposed by Chazan and Miranker in 1969, but with the choice of components and delays being based on probability distributions. We give sufficient conditions and necessary conditions for the expected value of the error to converge to zero. In addition we give sufficient conditions for the variance of the error to converge to zero. These conditions are all weaker than the strong condition of Chazan and Miranker. We also give numerical results of simulations illustrating the theoretical results.

Vertical slender jets with surface tension

The shape of a vertical slender jet of fluid falling steadily under the force of gravity is studied. The problem is formulated as a nonlinear free boundary-value problem for the potential. Surface-tension effects are included and studied. The use of perturbation expansions results in a system of equations that can be solved by an efficient numerical procedure. Computations were made for jets issuing from three different orifice shapes, which were an ellipse, a square, and an equilateral triangle. Computational results are presented illustrating the effects of different values for the surfacetension coefficient on the shape of the jet and the periodic nature of the cross-sectional shapes.