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Traveling-wave solutions to thin-film equations

Physical Review E 1993年第6期48卷 4423-4423页

作者： Stefanella Boatto Leo P. Kadanoff Piero Olla Computational and Applied Mathematics Program Ryerson Laboratory The University of Chicago Chicago Illinois 60637

Thin films can be effectively described by the lubrication approximation, in which the equation of motion is ht+(hnhxxx)x=0. Here h is a necessarily positive quantity which represents the height or thickness of the film. Different values of n, especially 1, 2, and 3 correspond to different physical situations. This equation permits solutions in the form of traveling disturbances with a fixed form. If u is the propagation velocity, the resulting equation for the disturbance is uhx=(hnhxxx)x. Here, quantitative and qualitative solutions to the equation are presented. The study has been limited to the intervals in x where the solutions are positive. It is found that transitions between different qualitative behaviors occur at n=3, 2, 3/2, and 1/2. For example, if u is not zero, solitonlike solutions defined on a finite interval are only possible for n<3. More specific results can be obtained. In the case in which the velocity is zero, solitons occur for n<2. For n=1, the region 3/21/2, single-minimum solutions diverging at ±∞ are possible. The generic solution, present for all positive values of n, is a receding front, which diverges at finite x for n<0.

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Heterogeneous Multiscale Methods: A Review

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Communications in computational Physics 2007年第3期2卷 367-450页

作者： Weinan E Bjorn Engquist Xiantao Li Weiqing Ren Eric Vanden-Eijnden Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Department of Mathematics The University of Texas at AustinAustinTX 78712USA Department of Mathematics Pennsylvania State UniversityUniversity ParkPA 16802USA Department of Mathematics Courant Institute of Mathematical SciencesNew York UniversityNew YorkNY 10012USA.

This paper gives a systematic introduction to HMM,the heterogeneous multiscale methods,including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular *** is illustrated by examples from several application areas,including complex fluids,micro-fluidics,solids,interface problems,stochastic problems,and statistically self-similar *** is given to the technical tools,such as the various constrained molecular dynamics,that have been developed,in order to apply HMM to these *** of mathematical results on the error analysis of HMM are *** review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.

关键词： Multi-scale modeling heterogeneous multi-scale method multi-physics models constrained micro-scale solver data estimation.

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Translation of a sphere towards a rigid plane in an Oldroyd-B fluid

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Physical Review Fluids 2024年第8期9卷 083303页

作者： Tachin Ruangkriengsin Rodolfo Brandão Bimalendu Mahapatra Evgeniy Boyko Howard A. Stone Program in Applied and Computational Mathematics Department of Mechanical and Aerospace Engineering Faculty of Mechanical Engineering

We analyze the low-Reynolds-number translation of a sphere towards or away from a rigid plane in an Oldroyd-B fluid under two scenarios: prescribing the sphere's translational velocity, and prescribing the force on the sphere. Leveraging the lubrication approximation and a perturbation expansion in powers of the Deborah number, we develop a comprehensive theoretical analysis that yields analytical approximations for velocity fields, pressures, and forces acting on the sphere. Our framework aids in understanding temporal microstructural changes as the particle-wall gap evolves over time. In particular, we show that alterations in the polymer conformation tensor in response to geometric changes induce additional forces on the sphere. For cases with prescribed velocity, we present a theoretical approach for calculating resistive forces at any order in the Deborah number and utilize a reciprocal theorem to obtain higher-order corrections based on velocity fields in the previous orders. When the sphere translates with a constant velocity, the fluid viscoelasticity decreases the resistive force at the first order. However, at the second-order correction, the direction of the sphere's movement determines whether viscoelasticity increases or decreases the resistive force. For cases with prescribed force, we show that understanding the influence of viscoelasticity on the sphere's translational velocity necessitates a more intricate analysis even at low Deborah numbers. Specifically, we introduce an ansatz for constant force scenarios, and we derive solution forms for general prescribed forces using the method of multiple scales. We find that when a sphere undergoes sedimentation due to its own weight, the fluid viscoelasticity results in a slower settling process, reducing the leading-order sedimentation rate.

关键词： Complex fluids Low Reynolds number flows Lubrication theory Polymer conformation changes Rheology

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The replicator dynamics for multilevel selection in evolutionary games

arXiv

引用

arXiv 2018年

作者： Cooney, Daniel B. Program in Applied and Computational Mathematics Princeton University

We consider a stochastic model for evolution of group-structured populations in which interactions between group members correspond to the Prisoner’s Dilemma or the Hawk-Dove game. Selection operates at two organization levels: individuals compete with peer group members based on individual payoff, while groups also compete with other groups based on average payoff of group members. In the Prisoner’s Dilemma, this creates a tension between the two levels of selection, as defectors are favored at the individual level, whereas groups with at least some cooperators outperform groups of defectors at the between-group level. In the limit of infinite group size and infinite number of groups, we derive a non-local PDE that describes the probability distribution of group compositions in the population. For special families of payoff matrices, we characterize the long-time behavior of solutions of our equation, finding a threshold intensity of between-group selection required to sustain density steady-states and the survival of cooperation. When all-cooperator groups are most fit, the average and most abundant group compositions at steady-state range from featuring all-defector groups when individual-level selection dominates to featuring all-cooperator groups when group-level selection dominates. When the most fit groups have a mix of cooperators and defectors, then the average and most abundant group compositions always feature a smaller fraction of cooperators than required for the optimal mix, even in the limit where group-level selection is infinitely stronger than individual-level selection. In such cases, the conflict between the two levels of selection cannot be decoupled, and cooperation cannot be sustained at all in the case where between-group competition favors an even mix of cooperators and *** Codes 92D25, 91A22, 35F20 Copyright © 2018, The Authors. All rights reserved.

关键词： Probability distributions

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Analysis of Multilevel Replicator Dynamics for General Two-Strategy Social Dilemmas

arXiv

引用

arXiv 2019年

作者： Cooney, Daniel B. Program in Applied and Computational Mathematics Princeton University

Here we consider a game-theoretic model of multilevel selection in which individuals compete based on their payoff and groups also compete based on the average payoff of group members. Our focus is on multilevel social dilemmas: games in which individuals are best off cheating, while groups of individuals do best when composed of many cooperators. We analyze the dynamics of the two-level replicator dynamics, a nonlocal hyperbolic PDE describing deterministic birth-death dynamics for both individuals and groups. While past work on such multilevel dynamics has restricted attention to scenarios with exactly solvable within-group dynamics, we use comparison principles and an invariant property of the tail of the population distribution to extend our analysis to all possible two-player, two-strategy social dilemmas. In the Stag-Hunt and similar games with coordination thresholds, we show that any amount of between-group competition allows for fixation of cooperation in the population. For the Prisoners’ Dilemma and Hawk-Dove game, we characterize the threshold level of between-group selection dividing a regime in which the population converges to a delta function at the equilibrium of the within-group dynamics from a regime in which between-group competition facilitates the existence of steady-state densities supporting greater levels of cooperation. In particular, we see that the threshold selection strength and average payoff at steady state depend on a tug-of-war between the individual-level incentive to be a defector in a many-cooperator group and the group-level incentive to have many cooperators over many defectors. We also find that lower-level selection casts a long shadow: if groups are best off with a mix of cooperators and defectors, then there will always be fewer cooperators than optimal at steady state, even in the limit of infinitely strong competition between groups. Copyright © 2019, The Authors. All rights reserved.

关键词： Dynamics

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Fire sale in financial networks

Fire sale in financial networks

引用

Annual Conference on Information Sciences and Systems (CISS)

作者： Haoshu Tian E Weinan Program in Applied and Computational Mathematics Princeton University

The default of one bank can cause other banks to default through two channels: financial contagion in the inter-bank liability network and fire sale in the asset selling market. When the defaulted bank cannot fully pay its debt, the loss is transmitted to other banks. When banks rush to sell the same asset simultaneously, they may fall into a Nash equilibrium in which banks compete for liquidity and sell their assets at an artificially low price. In this paper, a model that incorporates these two channels is developed and analyzed theoretically. An algorithm for finding the state in which both the inter-bank liability network and the market are in equilibrium is proposed and tested.

关键词： Vectors Nash equilibrium Fires Educational institutions Propagation losses Macroeconomics

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Numerical Algorithms for the Computation of Generalized Prolate Spheroidal Functions

arXiv

引用

arXiv 2017年

作者： Lederman, Roy R. Program in Applied and Computational Mathematics Princeton University

Generalized Prolate Spheroidal Functions (GPSF) are the eigenfunctions of the truncated Fourier transform, restricted to D-dimensional balls in the spatial domain and frequency domain. Despite their useful properties in many applications, GPSFs are often replaced by crude approximations. The purpose of this paper is to review the elements of computing GPSFs and associated eigenvalues. This paper is accompanied by open-source code.42A38, 65T99, 33F05, 65D20 Copyright © 2017, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

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Large‐scale structure generation by anisotropic small‐scale flows

The Physics of Fluids

引用

The Physics of Fluids 1987年第5期30卷 1272-1277页

作者： Victor Yakhot Richard Pelz Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544

Direct numerical simulation techniques are used to study the effect of spontaneous symmetry breaking leading to the phenomenon of large‐scale secondary flow generation by anisotropic small‐scale flow. It is shown that anisotropy is a major factor that determines the ability of the small‐scale flow to generate the inverse energy cascade, that the large‐scale secondary flow is strongly anisotropic—almost unidirectional, and that the helical property of the primary flow is irrelevent. The pairing instability and long‐wave modulation of the subharmonic are important in the process of generation of the large‐scale structure.

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Two shannon-type problems on secure multi-party computations

Two shannon-type problems on secure multi-party computations

引用

Annual Allerton Conference on Communication, Control, and Computing

作者： Eun Jee Lee Emmanuel Abbe Program in Applied and Computational Mathematics Princeton University

ISBN: (纸本)9781479980109

In secure multi-party computations (SMC), parties wish to compute a function on their private data without revealing more information about their data than what the function reveals. In this paper, we investigate two Shannon-type questions on this problem. We first consider the traditional one-shot model for SMC which does not assume a probabilistic prior on the data. In this model, private communication and randomness are the key enablers to secure computing, and we investigate a notion of randomness cost and capacity. We then move to a probabilistic model for the data, and propose a Shannon model for discrete memoryless SMC. In this model, correlations among data are the key enablers for secure computing, and we investigate a notion of dependency which permits the secure computation of a function. While the models and questions are general, this paper focuses on summation functions and relies on polar code constructions.

关键词： Protocols computational modeling Security Frequency modulation Data models Vectors Random variables

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Numerical computation of H^∞ optimal performance

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Journal of Scientific Computing 1992年第4期7卷 289-311页

作者： Yang, Hong Orszag, J.Michael Program in Applied and Computational Mathematics Princeton University Princeton 08544 New Jersey United States Department of Economics University of Michigan Ann Arbor 48109 Michigan United States

We present new algorithms for computing the H∞ optimal performance for a class of single-input/single-output (SISO) infinite-dimensional systems. The algorithms here only require use of one or two fast Fourier transf... 详细信息

We present new algorithms for computing the H^∞ optimal performance for a class of single-input/single-output (SISO) infinite-dimensional systems. The algorithms here only require use of one or two fast Fourier transforms (FFT) and Cholesky decompositions;hence the algorithms are particularly simple and easy to implement. Numerical examples show that the algorithms are stable and efficient and converge rapidly. The method has wide applications including to the H^∞ optimal control of distributed parameter systems. We illustrate the technique with applications to some delay problems and a partial differential equation (PDE) model. The algorithms we present are also an attractive approach to the solution of high-order finite-dimensional models for which use of state space methods would present computational difficulties. © 1992 Plenum Publishing Corporation.

关键词： H∞ control infinite-dimensional systems Krein space optimal performance

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